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Properties

Label 2-28e2-112.93-c1-0-5
Degree $2$
Conductor $784$
Sign $0.0446 - 0.999i$
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  + (−1.19 − 0.757i)2-s + (−0.538 + 2.01i)3-s + (0.851 + 1.80i)4-s + (−0.129 − 0.483i)5-s + (2.16 − 1.99i)6-s + (0.353 − 2.80i)8-s + (−1.15 − 0.667i)9-s + (−0.211 + 0.675i)10-s + (−0.456 − 0.122i)11-s + (−4.09 + 0.737i)12-s + (3.17 + 3.17i)13-s + 1.04·15-s + (−2.54 + 3.08i)16-s + (0.646 + 1.11i)17-s + (0.874 + 1.67i)18-s + (3.59 − 0.963i)19-s + ⋯
L(s)  = 1  + (−0.844 − 0.535i)2-s + (−0.311 + 1.16i)3-s + (0.425 + 0.904i)4-s + (−0.0579 − 0.216i)5-s + (0.884 − 0.813i)6-s + (0.125 − 0.992i)8-s + (−0.385 − 0.222i)9-s + (−0.0669 + 0.213i)10-s + (−0.137 − 0.0368i)11-s + (−1.18 + 0.213i)12-s + (0.881 + 0.881i)13-s + 0.269·15-s + (−0.637 + 0.770i)16-s + (0.156 + 0.271i)17-s + (0.206 + 0.394i)18-s + (0.824 − 0.221i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0446 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0446 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.0446 - 0.999i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.0446 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.628130 + 0.600692i\)
\(L(\frac12)\) \(\approx\) \(0.628130 + 0.600692i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 0.757i)T \)
7 \( 1 \)
good3 \( 1 + (0.538 - 2.01i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.129 + 0.483i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.456 + 0.122i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.17 - 3.17i)T + 13iT^{2} \)
17 \( 1 + (-0.646 - 1.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.59 + 0.963i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.26 - 1.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.98 + 6.98i)T + 29iT^{2} \)
31 \( 1 + (-4.17 - 7.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.21 + 4.53i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 9.93iT - 41T^{2} \)
43 \( 1 + (7.61 - 7.61i)T - 43iT^{2} \)
47 \( 1 + (2.29 - 3.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.38 - 1.44i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (8.73 + 2.34i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.94 + 1.59i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.65 - 13.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.62iT - 71T^{2} \)
73 \( 1 + (0.482 - 0.278i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.744 + 1.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.47 + 1.47i)T + 83iT^{2} \)
89 \( 1 + (10.6 + 6.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.50T + 97T^{2} \)
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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

−10.35859636545511697840082139426, −9.731566786447859709525118018275, −9.048881700468816815353684726355, −8.273726448113029277724791141857, −7.21180542438382567281126869445, −6.14546452590815363252473347591, −4.85926717613368206838313402714, −4.01372978536938192577354941722, −3.04237953624847186903175928855, −1.38089991177625148258270406978, 0.67410960467706994013364565261, 1.80144837465464268800172351550, 3.28342878914209987970077287260, 5.20731716786271705915322890624, 5.91495884451155029177692046583, 6.89605619975541914126984064558, 7.37886393773958078080544440510, 8.211765132633910968573451560758, 9.036847192158310622761254752176, 10.08294549217475066196958920564

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