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Properties

Label 2-28e2-112.37-c1-0-0
Degree $2$
Conductor $784$
Sign $-0.326 - 0.945i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.805 − 1.16i)2-s + (−1.16 − 0.312i)3-s + (−0.703 + 1.87i)4-s + (−0.980 + 0.262i)5-s + (0.575 + 1.60i)6-s + (2.74 − 0.690i)8-s + (−1.33 − 0.772i)9-s + (1.09 + 0.928i)10-s + (0.635 − 2.36i)11-s + (1.40 − 1.96i)12-s + (2.65 − 2.65i)13-s + 1.22·15-s + (−3.01 − 2.63i)16-s + (0.509 + 0.881i)17-s + (0.179 + 2.17i)18-s + (0.0250 + 0.0936i)19-s + ⋯
L(s)  = 1  + (−0.569 − 0.822i)2-s + (−0.672 − 0.180i)3-s + (−0.351 + 0.936i)4-s + (−0.438 + 0.117i)5-s + (0.234 + 0.655i)6-s + (0.969 − 0.243i)8-s + (−0.445 − 0.257i)9-s + (0.346 + 0.293i)10-s + (0.191 − 0.714i)11-s + (0.405 − 0.566i)12-s + (0.737 − 0.737i)13-s + 0.316·15-s + (−0.752 − 0.658i)16-s + (0.123 + 0.213i)17-s + (0.0422 + 0.513i)18-s + (0.00575 + 0.0214i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.326 - 0.945i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0105914 + 0.0148667i\)
\(L(\frac12)\) \(\approx\) \(0.0105914 + 0.0148667i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.805 + 1.16i)T \)
7 \( 1 \)
good3 \( 1 + (1.16 + 0.312i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.980 - 0.262i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.635 + 2.36i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-2.65 + 2.65i)T - 13iT^{2} \)
17 \( 1 + (-0.509 - 0.881i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0250 - 0.0936i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.67 - 0.965i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.05 - 5.05i)T - 29iT^{2} \)
31 \( 1 + (4.28 + 7.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.71 - 2.06i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 8.51iT - 41T^{2} \)
43 \( 1 + (4.47 + 4.47i)T + 43iT^{2} \)
47 \( 1 + (6.02 - 10.4i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.381 + 1.42i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.86 + 6.96i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.18 - 4.42i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (3.38 + 0.907i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.43iT - 71T^{2} \)
73 \( 1 + (7.34 - 4.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.433 - 0.751i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.44 + 5.44i)T - 83iT^{2} \)
89 \( 1 + (3.93 + 2.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.87862874306454625205285122878, −9.800242558418112417598557046880, −8.866726882956889824655167638228, −8.179403440306353653278777765984, −7.27878056787648673617700691464, −6.14762854919008462653095098724, −5.24078352419387244361719991918, −3.76111984737125265388585977735, −3.13033552032316573837337603777, −1.37883642741233389453418704781, 0.01290055256965017681750733984, 1.82550020239111226494369938112, 3.87037400696283994728733335288, 4.89415910367500808193268779536, 5.66466925787661586158746284706, 6.62918641865333032006545454776, 7.35362302118054798496411626717, 8.387176299381428843227397099976, 9.014749054017401964321967025285, 10.00303370749007450292725622306

Graph of the $Z$-function along the critical line