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Properties

Label 2-28e2-112.37-c1-0-75
Degree $2$
Conductor $784$
Sign $-0.953 - 0.301i$
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  + (0.945 − 1.05i)2-s + (0.0543 + 0.0145i)3-s + (−0.212 − 1.98i)4-s + (−1.25 + 0.337i)5-s + (0.0667 − 0.0434i)6-s + (−2.29 − 1.65i)8-s + (−2.59 − 1.49i)9-s + (−0.834 + 1.64i)10-s + (−0.402 + 1.50i)11-s + (0.0174 − 0.111i)12-s + (−1.59 + 1.59i)13-s − 0.0733·15-s + (−3.90 + 0.845i)16-s + (−1.46 − 2.54i)17-s + (−4.02 + 1.31i)18-s + (−2.05 − 7.65i)19-s + ⋯
L(s)  = 1  + (0.668 − 0.743i)2-s + (0.0313 + 0.00841i)3-s + (−0.106 − 0.994i)4-s + (−0.562 + 0.150i)5-s + (0.0272 − 0.0177i)6-s + (−0.810 − 0.585i)8-s + (−0.865 − 0.499i)9-s + (−0.263 + 0.519i)10-s + (−0.121 + 0.453i)11-s + (0.00502 − 0.0321i)12-s + (−0.442 + 0.442i)13-s − 0.0189·15-s + (−0.977 + 0.211i)16-s + (−0.356 − 0.617i)17-s + (−0.949 + 0.309i)18-s + (−0.470 − 1.75i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.953 - 0.301i$
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: \((2,\ 784,\ (\ :1/2),\ -0.953 - 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121853 + 0.788850i\)
\(L(\frac12)\) \(\approx\) \(0.121853 + 0.788850i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.945 + 1.05i)T \)
7 \( 1 \)
good3 \( 1 + (-0.0543 - 0.0145i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (1.25 - 0.337i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.402 - 1.50i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.59 - 1.59i)T - 13iT^{2} \)
17 \( 1 + (1.46 + 2.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.05 + 7.65i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.91 + 2.26i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.06 + 2.06i)T - 29iT^{2} \)
31 \( 1 + (-3.14 - 5.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.24 + 1.40i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 + (1.99 + 1.99i)T + 43iT^{2} \)
47 \( 1 + (0.979 - 1.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.00 + 11.2i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.793 - 2.96i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.60 + 9.72i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.11 + 0.566i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.26iT - 71T^{2} \)
73 \( 1 + (-12.2 + 7.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.961 - 1.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.82 - 8.82i)T - 83iT^{2} \)
89 \( 1 + (11.5 + 6.66i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.69T + 97T^{2} \)
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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

−9.868173082597651830952407901986, −9.216962913627056113605866517316, −8.257754915971648528189637248506, −6.96181716339020445344130002944, −6.28595513059481707957260435263, −5.00485892266625411723617748470, −4.33861831374481154838763554329, −3.14098228685496486096967469305, −2.27262953355154345643185299693, −0.29610624484203724110669921351, 2.40819826908824492520078146231, 3.62686937100004808972388798647, 4.42241736680941975563063134957, 5.70158821059930017111707474010, 6.05389104630848663494611231123, 7.46415661670822526048346011655, 8.146895882838249423899908616460, 8.533790650665346948911235026408, 9.902181977382474885773655486837, 10.90584102654231504980563118617

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