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Properties

Label 2-28e2-112.107-c0-0-0
Degree $2$
Conductor $784$
Sign $0.496 - 0.868i$
Analytic cond. $0.391266$
Root an. cond. $0.625513$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (0.866 + 0.5i)9-s + (−1.36 − 0.366i)11-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + (−0.999 − i)22-s + (1 − 1.73i)23-s + (−0.866 + 0.5i)25-s + (1 − i)29-s + (−0.866 + 0.499i)32-s + 0.999i·36-s + (−1.36 + 0.366i)37-s + (−1 + i)43-s + (−0.366 − 1.36i)44-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (0.866 + 0.5i)9-s + (−1.36 − 0.366i)11-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + (−0.999 − i)22-s + (1 − 1.73i)23-s + (−0.866 + 0.5i)25-s + (1 − i)29-s + (−0.866 + 0.499i)32-s + 0.999i·36-s + (−1.36 + 0.366i)37-s + (−1 + i)43-s + (−0.366 − 1.36i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.496 - 0.868i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :0),\ 0.496 - 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.541334090\)
\(L(\frac12)\) \(\approx\) \(1.541334090\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T^{2} \)
5 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1 + i)T - iT^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.66902795580643181635394708897, −10.02910475939848803816805734859, −8.564371068402713343074600174251, −7.962348204137922259291116922026, −7.09698757218468603499530286888, −6.25110118175128429337876503529, −5.11964682627567840236611993304, −4.58336630805791750171568083605, −3.27536228670837657177471745648, −2.21345868600080706104598064328, 1.57334716477303286272011606463, 2.89195887764347738093865720242, 3.88932592050006074511414690820, 4.96408329511130940346901685193, 5.63650562291649736988102925566, 6.90661095712952450031109156460, 7.44555998490019468433815833198, 8.816207712265261560324684130014, 9.936162844383575306814909795876, 10.30681331664546908427426373300

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