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Properties

Label 2-28e2-112.93-c1-0-57
Degree $2$
Conductor $784$
Sign $-0.995 + 0.0985i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.995 + 0.0985i$
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: \((2,\ 784,\ (\ :1/2),\ -0.995 + 0.0985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0388128 - 0.785859i\)
\(L(\frac12)\) \(\approx\) \(0.0388128 - 0.785859i\)
\(L(\frac{3}{2})\) 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

−9.921699307411429560525020840487, −8.957309986467319430926586408572, −8.088386048170217924472312428181, −7.43905183132942491576339400964, −6.86357791434176392038703765934, −5.66838922346641709357053931807, −4.06094721858807559677767646921, −2.68856944832679834066361625408, −2.03107125949675478015700644034, −0.48022993587207218798027986431, 1.77876815121105147817412983607, 3.27852992967650961517019080933, 4.68939818413969546908354680437, 5.16052430974849500717201801267, 6.60720538090427867554578019784, 7.23733998995208506344220128367, 8.526325613066609276497986858300, 9.049638043625384682982389206545, 9.634247265512965013380114819255, 10.56065892272198161549747505753

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