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Properties

Label 2-325-13.10-c1-0-11
Degree 22
Conductor 325325
Sign 0.964−0.265i0.964 - 0.265i
Analytic cond. 2.595132.59513
Root an. cond. 1.610941.61094
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  + (1.5 − 0.866i)2-s + (1 + 1.73i)3-s + (0.5 − 0.866i)4-s + (3 + 1.73i)6-s + 1.73i·8-s + (−0.499 + 0.866i)9-s + 2·12-s + (2.5 − 2.59i)13-s + (2.49 + 4.33i)16-s + (−1.5 + 2.59i)17-s + 1.73i·18-s + (−3 − 1.73i)19-s + (−3 − 5.19i)23-s + (−2.99 + 1.73i)24-s + (1.5 − 6.06i)26-s + 4.00·27-s + ⋯
L(s)  = 1  + (1.06 − 0.612i)2-s + (0.577 + 0.999i)3-s + (0.250 − 0.433i)4-s + (1.22 + 0.707i)6-s + 0.612i·8-s + (−0.166 + 0.288i)9-s + 0.577·12-s + (0.693 − 0.720i)13-s + (0.624 + 1.08i)16-s + (−0.363 + 0.630i)17-s + 0.408i·18-s + (−0.688 − 0.397i)19-s + (−0.625 − 1.08i)23-s + (−0.612 + 0.353i)24-s + (0.294 − 1.18i)26-s + 0.769·27-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((0.964−0.265i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+1/2) L(s)=((0.964−0.265i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: 0.964−0.265i0.964 - 0.265i
Analytic conductor: 2.595132.59513
Root analytic conductor: 1.610941.61094
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ325(101,⋅)\chi_{325} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 325, ( :1/2), 0.964−0.265i)(2,\ 325,\ (\ :1/2),\ 0.964 - 0.265i)

Particular Values

L(1)L(1) ≈\approx 2.53310+0.341754i2.53310 + 0.341754i
L(12)L(\frac12) ≈\approx 2.53310+0.341754i2.53310 + 0.341754i
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)
bad5 1 1
13 1+(−2.5+2.59i)T 1 + (-2.5 + 2.59i)T
good2 1+(−1.5+0.866i)T+(1−1.73i)T2 1 + (-1.5 + 0.866i)T + (1 - 1.73i)T^{2}
3 1+(−1−1.73i)T+(−1.5+2.59i)T2 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2}
7 1+(3.5+6.06i)T2 1 + (3.5 + 6.06i)T^{2}
11 1+(5.5−9.52i)T2 1 + (5.5 - 9.52i)T^{2}
17 1+(1.5−2.59i)T+(−8.5−14.7i)T2 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(3+1.73i)T+(9.5+16.4i)T2 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2}
23 1+(3+5.19i)T+(−11.5+19.9i)T2 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.5+2.59i)T+(−14.5+25.1i)T2 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2}
31 1+3.46iT−31T2 1 + 3.46iT - 31T^{2}
37 1+(7.5−4.33i)T+(18.5−32.0i)T2 1 + (7.5 - 4.33i)T + (18.5 - 32.0i)T^{2}
41 1+(4.5−2.59i)T+(20.5−35.5i)T2 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2}
43 1+(−4+6.92i)T+(−21.5−37.2i)T2 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2}
47 1+3.46iT−47T2 1 + 3.46iT - 47T^{2}
53 1−3T+53T2 1 - 3T + 53T^{2}
59 1+(−6−3.46i)T+(29.5+51.0i)T2 1 + (-6 - 3.46i)T + (29.5 + 51.0i)T^{2}
61 1+(0.5−0.866i)T+(−30.5−52.8i)T2 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2}
67 1+(3−1.73i)T+(33.5−58.0i)T2 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2}
71 1+(−3−1.73i)T+(35.5+61.4i)T2 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2}
73 1−1.73iT−73T2 1 - 1.73iT - 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−13.8iT−83T2 1 - 13.8iT - 83T^{2}
89 1+(6−3.46i)T+(44.5−77.0i)T2 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2}
97 1+(6+3.46i)T+(48.5+84.0i)T2 1 + (6 + 3.46i)T + (48.5 + 84.0i)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

−11.71670396963559990504457926355, −10.68006628426900369941810750173, −10.15538143203771762204339424550, −8.765344190845017559391492768609, −8.260113470217641071535381697909, −6.47138951629164004319458304994, −5.27685297911296615657841987840, −4.19843180921805728322683579281, −3.59011244658650847893802284355, −2.36543313134968192161402685996, 1.71764745575139670633268757715, 3.36148963433361345498900085481, 4.53821836070801265433574188228, 5.77398399297014933904924571342, 6.73250033221242312623278317749, 7.39633874029440911504684500765, 8.487785112214251969283663091759, 9.512541038628644593229302144859, 10.82646376404256733503150508582, 12.03683157845317867412048565630

Graph of the ZZ-function along the critical line