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Properties

Label 2-28e2-112.93-c1-0-23
Degree $2$
Conductor $784$
Sign $-0.810 - 0.585i$
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  + (1.03 + 0.960i)2-s + (−0.219 + 0.819i)3-s + (0.155 + 1.99i)4-s + (0.356 + 1.33i)5-s + (−1.01 + 0.640i)6-s + (−1.75 + 2.21i)8-s + (1.97 + 1.14i)9-s + (−0.908 + 1.72i)10-s + (2.35 + 0.631i)11-s + (−1.66 − 0.310i)12-s + (1.90 + 1.90i)13-s − 1.16·15-s + (−3.95 + 0.621i)16-s + (−3.35 − 5.81i)17-s + (0.955 + 3.07i)18-s + (−4.02 + 1.07i)19-s + ⋯
L(s)  = 1  + (0.734 + 0.679i)2-s + (−0.126 + 0.473i)3-s + (0.0778 + 0.996i)4-s + (0.159 + 0.595i)5-s + (−0.414 + 0.261i)6-s + (−0.619 + 0.784i)8-s + (0.658 + 0.380i)9-s + (−0.287 + 0.545i)10-s + (0.710 + 0.190i)11-s + (−0.481 − 0.0895i)12-s + (0.528 + 0.528i)13-s − 0.302·15-s + (−0.987 + 0.155i)16-s + (−0.814 − 1.41i)17-s + (0.225 + 0.725i)18-s + (−0.924 + 0.247i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.810 - 0.585i$
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: \((2,\ 784,\ (\ :1/2),\ -0.810 - 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703221 + 2.17544i\)
\(L(\frac12)\) \(\approx\) \(0.703221 + 2.17544i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 - 0.960i)T \)
7 \( 1 \)
good3 \( 1 + (0.219 - 0.819i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.356 - 1.33i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.35 - 0.631i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.90 - 1.90i)T + 13iT^{2} \)
17 \( 1 + (3.35 + 5.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.02 - 1.07i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.58 - 2.64i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.03 + 3.03i)T + 29iT^{2} \)
31 \( 1 + (0.599 + 1.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.823 + 3.07i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 3.94iT - 41T^{2} \)
43 \( 1 + (7.02 - 7.02i)T - 43iT^{2} \)
47 \( 1 + (-1.53 + 2.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.12 - 1.10i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.78 + 1.81i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-13.2 + 3.54i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.30 + 4.85i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + (-8.94 + 5.16i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.03 - 3.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.17 - 9.17i)T + 83iT^{2} \)
89 \( 1 + (-14.6 - 8.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.51T + 97T^{2} \)
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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

−10.91344640268534170344499076135, −9.618412233416978804401510515490, −8.995565042237760312415432001175, −7.82709451301070520200479244660, −6.84516966999658116353747142751, −6.47582794687789103173190430475, −5.14459705331276084163081550923, −4.41486565664821288695749355847, −3.50692868987081293655360726143, −2.18176192036031720139862944210, 0.990655590714848387666652333781, 1.97570230516280259492108325187, 3.55128258311342954952733369463, 4.33743270991948230717807285810, 5.40687128708165953801757219222, 6.40538854858994215076984775042, 6.94020560158070270679503851172, 8.587844335057293640044720809770, 9.028755181427795861849990575709, 10.26166569155563443165773667997

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