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Properties

Label 2-15e2-225.106-c1-0-13
Degree 22
Conductor 225225
Sign 0.603−0.797i0.603 - 0.797i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.91 + 0.406i)2-s + (−0.0970 + 1.72i)3-s + (1.67 + 0.744i)4-s + (2.20 − 0.343i)5-s + (−0.889 + 3.27i)6-s + (0.690 − 1.19i)7-s + (−0.268 − 0.194i)8-s + (−2.98 − 0.335i)9-s + (4.36 + 0.240i)10-s + (−2.41 − 0.513i)11-s + (−1.44 + 2.81i)12-s + (−3.14 + 0.668i)13-s + (1.80 − 2.00i)14-s + (0.380 + 3.85i)15-s + (−2.88 − 3.20i)16-s + (3.02 + 2.19i)17-s + ⋯
L(s)  = 1  + (1.35 + 0.287i)2-s + (−0.0560 + 0.998i)3-s + (0.836 + 0.372i)4-s + (0.988 − 0.153i)5-s + (−0.363 + 1.33i)6-s + (0.260 − 0.451i)7-s + (−0.0948 − 0.0689i)8-s + (−0.993 − 0.111i)9-s + (1.38 + 0.0761i)10-s + (−0.728 − 0.154i)11-s + (−0.418 + 0.813i)12-s + (−0.872 + 0.185i)13-s + (0.483 − 0.536i)14-s + (0.0981 + 0.995i)15-s + (−0.721 − 0.800i)16-s + (0.734 + 0.533i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((0.603−0.797i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((0.603−0.797i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 0.603−0.797i0.603 - 0.797i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ225(106,⋅)\chi_{225} (106, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), 0.603−0.797i)(2,\ 225,\ (\ :1/2),\ 0.603 - 0.797i)

Particular Values

L(1)L(1) ≈\approx 2.14643+1.06709i2.14643 + 1.06709i
L(12)L(\frac12) ≈\approx 2.14643+1.06709i2.14643 + 1.06709i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(0.0970−1.72i)T 1 + (0.0970 - 1.72i)T
5 1+(−2.20+0.343i)T 1 + (-2.20 + 0.343i)T
good2 1+(−1.91−0.406i)T+(1.82+0.813i)T2 1 + (-1.91 - 0.406i)T + (1.82 + 0.813i)T^{2}
7 1+(−0.690+1.19i)T+(−3.5−6.06i)T2 1 + (-0.690 + 1.19i)T + (-3.5 - 6.06i)T^{2}
11 1+(2.41+0.513i)T+(10.0+4.47i)T2 1 + (2.41 + 0.513i)T + (10.0 + 4.47i)T^{2}
13 1+(3.14−0.668i)T+(11.8−5.28i)T2 1 + (3.14 - 0.668i)T + (11.8 - 5.28i)T^{2}
17 1+(−3.02−2.19i)T+(5.25+16.1i)T2 1 + (-3.02 - 2.19i)T + (5.25 + 16.1i)T^{2}
19 1+(−0.234−0.170i)T+(5.87+18.0i)T2 1 + (-0.234 - 0.170i)T + (5.87 + 18.0i)T^{2}
23 1+(−0.389+0.432i)T+(−2.40−22.8i)T2 1 + (-0.389 + 0.432i)T + (-2.40 - 22.8i)T^{2}
29 1+(0.389+3.71i)T+(−28.3+6.02i)T2 1 + (0.389 + 3.71i)T + (-28.3 + 6.02i)T^{2}
31 1+(0.895−8.51i)T+(−30.3−6.44i)T2 1 + (0.895 - 8.51i)T + (-30.3 - 6.44i)T^{2}
37 1+(−2.37+7.31i)T+(−29.9−21.7i)T2 1 + (-2.37 + 7.31i)T + (-29.9 - 21.7i)T^{2}
41 1+(6.21−1.32i)T+(37.4−16.6i)T2 1 + (6.21 - 1.32i)T + (37.4 - 16.6i)T^{2}
43 1+(3.20−5.54i)T+(−21.5−37.2i)T2 1 + (3.20 - 5.54i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.750+7.14i)T+(−45.9+9.77i)T2 1 + (0.750 + 7.14i)T + (-45.9 + 9.77i)T^{2}
53 1+(−6.69+4.86i)T+(16.3−50.4i)T2 1 + (-6.69 + 4.86i)T + (16.3 - 50.4i)T^{2}
59 1+(−1.90+0.404i)T+(53.8−23.9i)T2 1 + (-1.90 + 0.404i)T + (53.8 - 23.9i)T^{2}
61 1+(−12.8−2.72i)T+(55.7+24.8i)T2 1 + (-12.8 - 2.72i)T + (55.7 + 24.8i)T^{2}
67 1+(1.05−10.0i)T+(−65.5−13.9i)T2 1 + (1.05 - 10.0i)T + (-65.5 - 13.9i)T^{2}
71 1+(10.4−7.59i)T+(21.9−67.5i)T2 1 + (10.4 - 7.59i)T + (21.9 - 67.5i)T^{2}
73 1+(−1.44−4.43i)T+(−59.0+42.9i)T2 1 + (-1.44 - 4.43i)T + (-59.0 + 42.9i)T^{2}
79 1+(−1.66−15.8i)T+(−77.2+16.4i)T2 1 + (-1.66 - 15.8i)T + (-77.2 + 16.4i)T^{2}
83 1+(−0.184+0.0821i)T+(55.5−61.6i)T2 1 + (-0.184 + 0.0821i)T + (55.5 - 61.6i)T^{2}
89 1+(3.30+10.1i)T+(−72.0+52.3i)T2 1 + (3.30 + 10.1i)T + (-72.0 + 52.3i)T^{2}
97 1+(1.15+10.9i)T+(−94.8+20.1i)T2 1 + (1.15 + 10.9i)T + (-94.8 + 20.1i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.65149221140935411820033726797, −11.58864033738547427254897120465, −10.35704142006206804333987796276, −9.775509690257185270976570003058, −8.518368201964769148622414886118, −6.95241528947918410697528413127, −5.64096951527383129210664694203, −5.15024958358352213503980170990, −4.05111032356920334331507121341, −2.73000832022930014587681479074, 2.11550674966413175735722515466, 2.99189197328607221152975437309, 5.07078119251077265574879878628, 5.58786806444977730864720872370, 6.69721149062030205827385064698, 7.88275020311444815097754782723, 9.196585332403709450823209442374, 10.45877441917974704725006984354, 11.68552956458789431394096923732, 12.26748937941340603770096168566

Graph of the ZZ-function along the critical line