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Properties

Label 2-28e2-112.37-c1-0-50
Degree $2$
Conductor $784$
Sign $0.981 - 0.192i$
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  + (1.39 − 0.250i)2-s + (2.46 + 0.659i)3-s + (1.87 − 0.697i)4-s + (−1.87 + 0.502i)5-s + (3.58 + 0.300i)6-s + (2.43 − 1.44i)8-s + (3.02 + 1.74i)9-s + (−2.48 + 1.16i)10-s + (−1.37 + 5.11i)11-s + (5.07 − 0.481i)12-s + (3.61 − 3.61i)13-s − 4.94·15-s + (3.02 − 2.61i)16-s + (0.294 + 0.510i)17-s + (4.64 + 1.66i)18-s + (0.137 + 0.514i)19-s + ⋯
L(s)  = 1  + (0.984 − 0.177i)2-s + (1.42 + 0.380i)3-s + (0.937 − 0.348i)4-s + (−0.838 + 0.224i)5-s + (1.46 + 0.122i)6-s + (0.860 − 0.509i)8-s + (1.00 + 0.581i)9-s + (−0.785 + 0.369i)10-s + (−0.413 + 1.54i)11-s + (1.46 − 0.139i)12-s + (1.00 − 1.00i)13-s − 1.27·15-s + (0.756 − 0.654i)16-s + (0.0715 + 0.123i)17-s + (1.09 + 0.393i)18-s + (0.0316 + 0.118i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.981 - 0.192i$
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: \((2,\ 784,\ (\ :1/2),\ 0.981 - 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.94276 + 0.382694i\)
\(L(\frac12)\) \(\approx\) \(3.94276 + 0.382694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.250i)T \)
7 \( 1 \)
good3 \( 1 + (-2.46 - 0.659i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (1.87 - 0.502i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.37 - 5.11i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.61 + 3.61i)T - 13iT^{2} \)
17 \( 1 + (-0.294 - 0.510i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.137 - 0.514i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.378 - 0.218i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.68 + 2.68i)T - 29iT^{2} \)
31 \( 1 + (3.94 + 6.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.90 - 0.779i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.95iT - 41T^{2} \)
43 \( 1 + (7.67 + 7.67i)T + 43iT^{2} \)
47 \( 1 + (2.01 - 3.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.244 + 0.912i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.00753 - 0.0281i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.529 + 1.97i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (6.24 + 1.67i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.05iT - 71T^{2} \)
73 \( 1 + (6.85 - 3.95i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.53 - 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 + (-3.35 - 1.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−10.25492779940891482764296948188, −9.687096133396809370448391234711, −8.379107741383752561560440206583, −7.75556480353878932208987648531, −7.04947390463099851797697318774, −5.70496439945033673553481973777, −4.50229002607215306097234719478, −3.77610324140904468426528674648, −3.02730512428721683068163013770, −1.94249503808649394113380465517, 1.65399343396360614156325059324, 3.13526342009226950810439492787, 3.50538713985230669926411147348, 4.60002840375163351760317422663, 5.88485397876823072670061599060, 6.88893880083882151498889151691, 7.72030088889874380462530785490, 8.554579991809743842626937402803, 8.826351053505439525279992022034, 10.47960646740091692192755769741

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