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Properties

Label 2-1412-1412.387-c0-0-0
Degree $2$
Conductor $1412$
Sign $0.144 - 0.989i$
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  + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (1.83 + 0.682i)5-s + (0.959 − 0.281i)8-s + (0.755 + 0.654i)9-s + (−1.38 + 1.38i)10-s + (−0.418 − 0.559i)13-s + (−0.142 + 0.989i)16-s + (−1.25 + 0.368i)17-s + (−0.909 + 0.415i)18-s + (−0.682 − 1.83i)20-s + (2.13 + 1.84i)25-s + (0.682 − 0.148i)26-s + (1.29 − 1.49i)29-s + (−0.841 − 0.540i)32-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (1.83 + 0.682i)5-s + (0.959 − 0.281i)8-s + (0.755 + 0.654i)9-s + (−1.38 + 1.38i)10-s + (−0.418 − 0.559i)13-s + (−0.142 + 0.989i)16-s + (−1.25 + 0.368i)17-s + (−0.909 + 0.415i)18-s + (−0.682 − 1.83i)20-s + (2.13 + 1.84i)25-s + (0.682 − 0.148i)26-s + (1.29 − 1.49i)29-s + (−0.841 − 0.540i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $0.144 - 0.989i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ 0.144 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.159786549\)
\(L(\frac12)\) \(\approx\) \(1.159786549\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
353 \( 1 + (0.959 + 0.281i)T \)
good3 \( 1 + (-0.755 - 0.654i)T^{2} \)
5 \( 1 + (-1.83 - 0.682i)T + (0.755 + 0.654i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.418 + 0.559i)T + (-0.281 + 0.959i)T^{2} \)
17 \( 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.959 + 0.281i)T^{2} \)
23 \( 1 + (0.415 + 0.909i)T^{2} \)
29 \( 1 + (-1.29 + 1.49i)T + (-0.142 - 0.989i)T^{2} \)
31 \( 1 + (0.909 + 0.415i)T^{2} \)
37 \( 1 + (1.54 + 0.841i)T + (0.540 + 0.841i)T^{2} \)
41 \( 1 + (1.03 + 1.61i)T + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.142 + 0.989i)T^{2} \)
53 \( 1 + (0.0498 - 0.133i)T + (-0.755 - 0.654i)T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.281 - 1.95i)T + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (0.989 + 0.142i)T^{2} \)
73 \( 1 + (-0.142 + 0.989i)T^{2} \)
79 \( 1 + (-0.755 + 0.654i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (1.71 - 0.936i)T + (0.540 - 0.841i)T^{2} \)
97 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
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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

−9.966148440433563810253361119340, −9.143726065127604118472661789827, −8.385250522515696380874979822067, −7.22807190214974211490583830669, −6.76689979944053253580598867107, −5.90252032156205445939146574799, −5.27051477800384565298147578389, −4.30164983610761613930684899763, −2.53081915813262818083575528975, −1.67912652368452641011649315106, 1.33898727833752210470641386358, 2.05938533962624588179913753855, 3.19821211033273145692121288940, 4.67667852364847450890091948671, 4.99517606896388735807342693579, 6.47625497723847101386207841075, 6.93904183547533082825050642322, 8.562715087827933193153705059641, 8.878705412900829918979529997906, 9.857273708631799043109784292969

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