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Properties

Label 2-28e2-16.13-c1-0-71
Degree $2$
Conductor $784$
Sign $-0.0234 + 0.999i$
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  + (1.35 − 0.402i)2-s + (−0.631 − 0.631i)3-s + (1.67 − 1.09i)4-s + (2.34 − 2.34i)5-s + (−1.10 − 0.601i)6-s + (1.83 − 2.15i)8-s − 2.20i·9-s + (2.23 − 4.11i)10-s + (−2.18 + 2.18i)11-s + (−1.74 − 0.368i)12-s + (4.03 + 4.03i)13-s − 2.95·15-s + (1.61 − 3.65i)16-s − 0.347·17-s + (−0.886 − 2.98i)18-s + (−4.26 − 4.26i)19-s + ⋯
L(s)  = 1  + (0.958 − 0.284i)2-s + (−0.364 − 0.364i)3-s + (0.837 − 0.545i)4-s + (1.04 − 1.04i)5-s + (−0.453 − 0.245i)6-s + (0.647 − 0.761i)8-s − 0.734i·9-s + (0.706 − 1.30i)10-s + (−0.658 + 0.658i)11-s + (−0.504 − 0.106i)12-s + (1.11 + 1.11i)13-s − 0.763·15-s + (0.404 − 0.914i)16-s − 0.0843·17-s + (−0.209 − 0.704i)18-s + (−0.978 − 0.978i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0234 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0234 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.0234 + 0.999i$
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: \((2,\ 784,\ (\ :1/2),\ -0.0234 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03740 - 2.08579i\)
\(L(\frac12)\) \(\approx\) \(2.03740 - 2.08579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.402i)T \)
7 \( 1 \)
good3 \( 1 + (0.631 + 0.631i)T + 3iT^{2} \)
5 \( 1 + (-2.34 + 2.34i)T - 5iT^{2} \)
11 \( 1 + (2.18 - 2.18i)T - 11iT^{2} \)
13 \( 1 + (-4.03 - 4.03i)T + 13iT^{2} \)
17 \( 1 + 0.347T + 17T^{2} \)
19 \( 1 + (4.26 + 4.26i)T + 19iT^{2} \)
23 \( 1 - 6.23iT - 23T^{2} \)
29 \( 1 + (-1.21 - 1.21i)T + 29iT^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 + (6.42 - 6.42i)T - 37iT^{2} \)
41 \( 1 - 2.68iT - 41T^{2} \)
43 \( 1 + (-4.05 + 4.05i)T - 43iT^{2} \)
47 \( 1 + 4.64T + 47T^{2} \)
53 \( 1 + (-8.44 + 8.44i)T - 53iT^{2} \)
59 \( 1 + (-5.17 + 5.17i)T - 59iT^{2} \)
61 \( 1 + (-0.00533 - 0.00533i)T + 61iT^{2} \)
67 \( 1 + (-3.02 - 3.02i)T + 67iT^{2} \)
71 \( 1 - 0.828iT - 71T^{2} \)
73 \( 1 - 6.25iT - 73T^{2} \)
79 \( 1 + 0.755T + 79T^{2} \)
83 \( 1 + (-3.66 - 3.66i)T + 83iT^{2} \)
89 \( 1 + 6.24iT - 89T^{2} \)
97 \( 1 + 2.18T + 97T^{2} \)
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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

−10.11350885603332964093709487380, −9.363400299746065035566307715616, −8.543588533880459901587720345479, −6.98684059878427044674195995117, −6.42638369464586253658678164184, −5.51748659836550032531049279132, −4.80076384645953109121216428942, −3.73673681009830920783211895451, −2.14676219387995226788234611697, −1.25346595960211204817691241437, 2.16094457043165057572197527955, 3.02835655042583883039801869893, 4.18190917082137366369724428253, 5.49956321292900100855978132293, 5.87247673566179829607648477401, 6.64539827918290919196057153754, 7.84304205150988439784718113786, 8.552602677526238912251163608042, 10.25286019905630705637163812552, 10.67727471388445205180575181610

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