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Properties

Label 2-28e2-16.13-c1-0-51
Degree $2$
Conductor $784$
Sign $0.989 + 0.142i$
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  + (0.857 + 1.12i)2-s + (−0.416 − 0.416i)3-s + (−0.529 + 1.92i)4-s + (1.13 − 1.13i)5-s + (0.111 − 0.826i)6-s + (−2.62 + 1.05i)8-s − 2.65i·9-s + (2.24 + 0.302i)10-s + (3.85 − 3.85i)11-s + (1.02 − 0.583i)12-s + (−4.66 − 4.66i)13-s − 0.943·15-s + (−3.43 − 2.04i)16-s + 5.33·17-s + (2.98 − 2.27i)18-s + (2.55 + 2.55i)19-s + ⋯
L(s)  = 1  + (0.606 + 0.795i)2-s + (−0.240 − 0.240i)3-s + (−0.264 + 0.964i)4-s + (0.506 − 0.506i)5-s + (0.0454 − 0.337i)6-s + (−0.927 + 0.374i)8-s − 0.884i·9-s + (0.709 + 0.0956i)10-s + (1.16 − 1.16i)11-s + (0.295 − 0.168i)12-s + (−1.29 − 1.29i)13-s − 0.243·15-s + (−0.859 − 0.510i)16-s + 1.29·17-s + (0.703 − 0.536i)18-s + (0.587 + 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.989 + 0.142i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.989 + 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98379 - 0.142446i\)
\(L(\frac12)\) \(\approx\) \(1.98379 - 0.142446i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.857 - 1.12i)T \)
7 \( 1 \)
good3 \( 1 + (0.416 + 0.416i)T + 3iT^{2} \)
5 \( 1 + (-1.13 + 1.13i)T - 5iT^{2} \)
11 \( 1 + (-3.85 + 3.85i)T - 11iT^{2} \)
13 \( 1 + (4.66 + 4.66i)T + 13iT^{2} \)
17 \( 1 - 5.33T + 17T^{2} \)
19 \( 1 + (-2.55 - 2.55i)T + 19iT^{2} \)
23 \( 1 + 2.60iT - 23T^{2} \)
29 \( 1 + (1.22 + 1.22i)T + 29iT^{2} \)
31 \( 1 - 0.833T + 31T^{2} \)
37 \( 1 + (4.42 - 4.42i)T - 37iT^{2} \)
41 \( 1 - 0.263iT - 41T^{2} \)
43 \( 1 + (-1.25 + 1.25i)T - 43iT^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + (-0.0476 + 0.0476i)T - 53iT^{2} \)
59 \( 1 + (-3.60 + 3.60i)T - 59iT^{2} \)
61 \( 1 + (4.46 + 4.46i)T + 61iT^{2} \)
67 \( 1 + (-9.50 - 9.50i)T + 67iT^{2} \)
71 \( 1 - 2.05iT - 71T^{2} \)
73 \( 1 - 5.48iT - 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 + (5.84 + 5.84i)T + 83iT^{2} \)
89 \( 1 - 6.32iT - 89T^{2} \)
97 \( 1 + 18.8T + 97T^{2} \)
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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

−10.02893403392449393643311056322, −9.307289847669562065404434550166, −8.436126474504863117224350337068, −7.56539858138328038728663307094, −6.64079227959161878827448727497, −5.67246211335548104324164916950, −5.35810316779159236962620701378, −3.85623597825475476239059292228, −3.02994039812847438531109751740, −0.913032474368678230306694346627, 1.69551579924970737754873145237, 2.55271419548687756985818221407, 3.97906835495047425285243016603, 4.79471581124356899091269679357, 5.60592127040235087117428154880, 6.77290307336528790657062052091, 7.43362219433981301026746815935, 9.176127272540091177245013883572, 9.690388800634393961809262859310, 10.28888814192041243256994282221

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