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Properties

Label 2-405-135.122-c1-0-6
Degree 22
Conductor 405405
Sign −0.331−0.943i-0.331 - 0.943i
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
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  + (2.18 + 1.52i)2-s + (1.73 + 4.77i)4-s + (−1.64 − 1.51i)5-s + (1.30 + 2.78i)7-s + (−2.12 + 7.92i)8-s + (−1.28 − 5.81i)10-s + (0.426 + 0.508i)11-s + (−0.269 − 0.384i)13-s + (−1.42 + 8.06i)14-s + (−8.94 + 7.50i)16-s + (−1.20 − 4.50i)17-s + (4.80 − 2.77i)19-s + (4.34 − 10.5i)20-s + (0.154 + 1.76i)22-s + (0.0602 + 0.0280i)23-s + ⋯
L(s)  = 1  + (1.54 + 1.07i)2-s + (0.869 + 2.38i)4-s + (−0.737 − 0.675i)5-s + (0.491 + 1.05i)7-s + (−0.750 + 2.80i)8-s + (−0.407 − 1.83i)10-s + (0.128 + 0.153i)11-s + (−0.0746 − 0.106i)13-s + (−0.380 + 2.15i)14-s + (−2.23 + 1.87i)16-s + (−0.292 − 1.09i)17-s + (1.10 − 0.637i)19-s + (0.972 − 2.34i)20-s + (0.0328 + 0.375i)22-s + (0.0125 + 0.00585i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.331−0.943i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((−0.331−0.943i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.331−0.943i-0.331 - 0.943i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(152,⋅)\chi_{405} (152, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), −0.331−0.943i)(2,\ 405,\ (\ :1/2),\ -0.331 - 0.943i)

Particular Values

L(1)L(1) ≈\approx 1.66186+2.34605i1.66186 + 2.34605i
L(12)L(\frac12) ≈\approx 1.66186+2.34605i1.66186 + 2.34605i
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 1
5 1+(1.64+1.51i)T 1 + (1.64 + 1.51i)T
good2 1+(−2.18−1.52i)T+(0.684+1.87i)T2 1 + (-2.18 - 1.52i)T + (0.684 + 1.87i)T^{2}
7 1+(−1.30−2.78i)T+(−4.49+5.36i)T2 1 + (-1.30 - 2.78i)T + (-4.49 + 5.36i)T^{2}
11 1+(−0.426−0.508i)T+(−1.91+10.8i)T2 1 + (-0.426 - 0.508i)T + (-1.91 + 10.8i)T^{2}
13 1+(0.269+0.384i)T+(−4.44+12.2i)T2 1 + (0.269 + 0.384i)T + (-4.44 + 12.2i)T^{2}
17 1+(1.20+4.50i)T+(−14.7+8.5i)T2 1 + (1.20 + 4.50i)T + (-14.7 + 8.5i)T^{2}
19 1+(−4.80+2.77i)T+(9.5−16.4i)T2 1 + (-4.80 + 2.77i)T + (9.5 - 16.4i)T^{2}
23 1+(−0.0602−0.0280i)T+(14.7+17.6i)T2 1 + (-0.0602 - 0.0280i)T + (14.7 + 17.6i)T^{2}
29 1+(0.434+2.46i)T+(−27.2+9.91i)T2 1 + (0.434 + 2.46i)T + (-27.2 + 9.91i)T^{2}
31 1+(−1.76+0.642i)T+(23.7−19.9i)T2 1 + (-1.76 + 0.642i)T + (23.7 - 19.9i)T^{2}
37 1+(2.44−0.656i)T+(32.0−18.5i)T2 1 + (2.44 - 0.656i)T + (32.0 - 18.5i)T^{2}
41 1+(1.62+0.286i)T+(38.5+14.0i)T2 1 + (1.62 + 0.286i)T + (38.5 + 14.0i)T^{2}
43 1+(−0.732+8.36i)T+(−42.3−7.46i)T2 1 + (-0.732 + 8.36i)T + (-42.3 - 7.46i)T^{2}
47 1+(−3.71+1.73i)T+(30.2−36.0i)T2 1 + (-3.71 + 1.73i)T + (30.2 - 36.0i)T^{2}
53 1+(7.52+7.52i)T+53iT2 1 + (7.52 + 7.52i)T + 53iT^{2}
59 1+(−3.49−2.93i)T+(10.2+58.1i)T2 1 + (-3.49 - 2.93i)T + (10.2 + 58.1i)T^{2}
61 1+(5.84+2.12i)T+(46.7+39.2i)T2 1 + (5.84 + 2.12i)T + (46.7 + 39.2i)T^{2}
67 1+(4.48−3.13i)T+(22.9−62.9i)T2 1 + (4.48 - 3.13i)T + (22.9 - 62.9i)T^{2}
71 1+(−5.33−3.07i)T+(35.5+61.4i)T2 1 + (-5.33 - 3.07i)T + (35.5 + 61.4i)T^{2}
73 1+(−13.6−3.64i)T+(63.2+36.5i)T2 1 + (-13.6 - 3.64i)T + (63.2 + 36.5i)T^{2}
79 1+(5.34−0.941i)T+(74.2−27.0i)T2 1 + (5.34 - 0.941i)T + (74.2 - 27.0i)T^{2}
83 1+(3.25−4.64i)T+(−28.3−77.9i)T2 1 + (3.25 - 4.64i)T + (-28.3 - 77.9i)T^{2}
89 1+(−3.08−5.33i)T+(−44.5+77.0i)T2 1 + (-3.08 - 5.33i)T + (-44.5 + 77.0i)T^{2}
97 1+(13.5+1.18i)T+(95.5+16.8i)T2 1 + (13.5 + 1.18i)T + (95.5 + 16.8i)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

−11.86996905399605425964482214695, −11.34548548163029308243137493582, −9.295116873170174149617365767046, −8.417627947138894565559342500776, −7.59332599156702575667137336630, −6.74414603358114757987878833911, −5.34791415791044157847094926186, −5.05767292715144592318576284610, −3.89789805854824542631945508793, −2.64690700998015598573477744532, 1.43163658792941009953552947446, 3.06092208227692981404136201494, 3.90154662152656348097049963388, 4.65283950859573844231999708278, 5.94977980629238301174746861706, 6.94723858858363522877646588867, 7.979216235079794447420594496673, 9.713475800888984100365190774942, 10.74738226297253276426358648490, 10.94651734947672233096711297708

Graph of the ZZ-function along the critical line