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Properties

Label 2-1412-1412.1055-c0-0-0
Degree $2$
Conductor $1412$
Sign $0.141 + 0.989i$
Analytic cond. $0.704679$
Root an. cond. $0.839452$
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  + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.203 − 0.936i)5-s + (0.142 + 0.989i)8-s + (−0.909 + 0.415i)9-s + (0.677 + 0.677i)10-s + (0.139 − 1.94i)13-s + (−0.654 − 0.755i)16-s + (0.118 + 0.822i)17-s + (0.540 − 0.841i)18-s + (−0.936 − 0.203i)20-s + (0.0739 − 0.0337i)25-s + (0.936 + 1.71i)26-s + (−0.627 − 1.37i)29-s + (0.959 + 0.281i)32-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.203 − 0.936i)5-s + (0.142 + 0.989i)8-s + (−0.909 + 0.415i)9-s + (0.677 + 0.677i)10-s + (0.139 − 1.94i)13-s + (−0.654 − 0.755i)16-s + (0.118 + 0.822i)17-s + (0.540 − 0.841i)18-s + (−0.936 − 0.203i)20-s + (0.0739 − 0.0337i)25-s + (0.936 + 1.71i)26-s + (−0.627 − 1.37i)29-s + (0.959 + 0.281i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $0.141 + 0.989i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ 0.141 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5173662667\)
\(L(\frac12)\) \(\approx\) \(0.5173662667\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 0.540i)T \)
353 \( 1 + (0.142 - 0.989i)T \)
good3 \( 1 + (0.909 - 0.415i)T^{2} \)
5 \( 1 + (0.203 + 0.936i)T + (-0.909 + 0.415i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (-0.139 + 1.94i)T + (-0.989 - 0.142i)T^{2} \)
17 \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \)
19 \( 1 + (-0.142 - 0.989i)T^{2} \)
23 \( 1 + (0.841 + 0.540i)T^{2} \)
29 \( 1 + (0.627 + 1.37i)T + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (-0.540 - 0.841i)T^{2} \)
37 \( 1 + (1.28 + 0.959i)T + (0.281 + 0.959i)T^{2} \)
41 \( 1 + (0.0801 + 0.273i)T + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (-0.654 - 0.755i)T^{2} \)
53 \( 1 + (0.682 - 0.148i)T + (0.909 - 0.415i)T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.989 + 1.14i)T + (-0.142 + 0.989i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (0.755 - 0.654i)T^{2} \)
73 \( 1 + (-0.654 - 0.755i)T^{2} \)
79 \( 1 + (0.909 + 0.415i)T^{2} \)
83 \( 1 + (0.959 + 0.281i)T^{2} \)
89 \( 1 + (-0.767 + 0.574i)T + (0.281 - 0.959i)T^{2} \)
97 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
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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

−9.361530861449420737055653528027, −8.542344126653749384971905463154, −8.128383798939942667274989966165, −7.50255443982111162974366258811, −6.15304478713553724671523141315, −5.56406351044480376067138879818, −4.88065304792817478567476621297, −3.39832597622781705336802896617, −2.07135298406014991774317380160, −0.55217582369366805732480787694, 1.67789766662466335833982943329, 2.88310792688989018040429264151, 3.51407027854897179306827154731, 4.70493717664749178035274872967, 6.18523493121385195198678253512, 6.91600104477889026525281324809, 7.42124554846666063725675989520, 8.680005869924840215947878551324, 9.053099791539085211240783921678, 9.872074689753604143265486916192

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