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Properties

Label 2-28e2-112.19-c1-0-53
Degree $2$
Conductor $784$
Sign $0.228 + 0.973i$
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  + (−1.41 + 0.105i)2-s + (0.551 − 2.05i)3-s + (1.97 − 0.297i)4-s + (3.47 − 0.929i)5-s + (−0.560 + 2.95i)6-s + (−2.75 + 0.627i)8-s + (−1.33 − 0.768i)9-s + (−4.79 + 1.67i)10-s + (−0.732 + 2.73i)11-s + (0.478 − 4.23i)12-s + (1.17 − 1.17i)13-s − 7.65i·15-s + (3.82 − 1.17i)16-s + (5.31 − 3.06i)17-s + (1.95 + 0.943i)18-s + (−1.33 + 0.358i)19-s + ⋯
L(s)  = 1  + (−0.997 + 0.0745i)2-s + (0.318 − 1.18i)3-s + (0.988 − 0.148i)4-s + (1.55 − 0.415i)5-s + (−0.228 + 1.20i)6-s + (−0.975 + 0.221i)8-s + (−0.443 − 0.256i)9-s + (−1.51 + 0.530i)10-s + (−0.220 + 0.823i)11-s + (0.138 − 1.22i)12-s + (0.326 − 0.326i)13-s − 1.97i·15-s + (0.955 − 0.294i)16-s + (1.28 − 0.743i)17-s + (0.461 + 0.222i)18-s + (−0.307 + 0.0822i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.228 + 0.973i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.228 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19150 - 0.943865i\)
\(L(\frac12)\) \(\approx\) \(1.19150 - 0.943865i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.105i)T \)
7 \( 1 \)
good3 \( 1 + (-0.551 + 2.05i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-3.47 + 0.929i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.732 - 2.73i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.17 + 1.17i)T - 13iT^{2} \)
17 \( 1 + (-5.31 + 3.06i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.33 - 0.358i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.103 - 0.179i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.46 + 3.46i)T + 29iT^{2} \)
31 \( 1 + (-2.87 - 4.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0935 + 0.349i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.75T + 41T^{2} \)
43 \( 1 + (-0.207 - 0.207i)T + 43iT^{2} \)
47 \( 1 + (5.46 - 9.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.82 - 2.63i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.77 + 0.743i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.74 + 10.2i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.33 + 0.358i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.09T + 71T^{2} \)
73 \( 1 + (1.36 + 2.36i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.55 + 5.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.27 - 2.27i)T + 83iT^{2} \)
89 \( 1 + (7.04 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.83iT - 97T^{2} \)
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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

−9.902807236982884823002279844587, −9.334761913313368376647159649368, −8.330148240396052100545338245454, −7.62928558195154027104507396355, −6.78655619995560113239585515023, −6.01840484035611697754016923032, −5.12417446529145712727704706846, −2.88114057189027002463481850451, −1.92958153407256390274080600615, −1.16430945934850964607271026388, 1.57217019775842444362459201332, 2.83634430414788495178721232735, 3.73468441789545909061517298987, 5.41438956791605798994406971054, 6.04138642303990279169732782203, 7.02964567261989517980069541946, 8.366924352742894022130167091305, 8.938606247672509426944042629564, 9.849402449903658421042956715607, 10.16891060197680263094220504081

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