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Properties

Label 2-1412-1412.515-c0-0-0
Degree $2$
Conductor $1412$
Sign $-0.833 - 0.552i$
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  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.133 + 1.86i)5-s + (−0.841 + 0.540i)8-s + (−0.989 + 0.142i)9-s + (−1.32 + 1.32i)10-s + (1.75 − 0.956i)13-s + (−0.959 − 0.281i)16-s + (0.239 − 0.153i)17-s + (−0.755 − 0.654i)18-s + (−1.86 − 0.133i)20-s + (−2.48 + 0.357i)25-s + (1.86 + 0.697i)26-s + (−0.0801 − 0.557i)29-s + (−0.415 − 0.909i)32-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.133 + 1.86i)5-s + (−0.841 + 0.540i)8-s + (−0.989 + 0.142i)9-s + (−1.32 + 1.32i)10-s + (1.75 − 0.956i)13-s + (−0.959 − 0.281i)16-s + (0.239 − 0.153i)17-s + (−0.755 − 0.654i)18-s + (−1.86 − 0.133i)20-s + (−2.48 + 0.357i)25-s + (1.86 + 0.697i)26-s + (−0.0801 − 0.557i)29-s + (−0.415 − 0.909i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $-0.833 - 0.552i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ -0.833 - 0.552i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.429068959\)
\(L(\frac12)\) \(\approx\) \(1.429068959\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
353 \( 1 + (-0.841 - 0.540i)T \)
good3 \( 1 + (0.989 - 0.142i)T^{2} \)
5 \( 1 + (-0.133 - 1.86i)T + (-0.989 + 0.142i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (0.654 + 0.755i)T^{2} \)
13 \( 1 + (-1.75 + 0.956i)T + (0.540 - 0.841i)T^{2} \)
17 \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (0.841 - 0.540i)T^{2} \)
23 \( 1 + (-0.654 + 0.755i)T^{2} \)
29 \( 1 + (0.0801 + 0.557i)T + (-0.959 + 0.281i)T^{2} \)
31 \( 1 + (0.755 - 0.654i)T^{2} \)
37 \( 1 + (1.90 + 0.415i)T + (0.909 + 0.415i)T^{2} \)
41 \( 1 + (-1.53 - 0.698i)T + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.959 - 0.281i)T^{2} \)
53 \( 1 + (-1.59 + 0.114i)T + (0.989 - 0.142i)T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-0.540 - 0.158i)T + (0.841 + 0.540i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (-0.281 + 0.959i)T^{2} \)
73 \( 1 + (-0.959 - 0.281i)T^{2} \)
79 \( 1 + (0.989 + 0.142i)T^{2} \)
83 \( 1 + (-0.415 - 0.909i)T^{2} \)
89 \( 1 + (-1.83 + 0.398i)T + (0.909 - 0.415i)T^{2} \)
97 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
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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

−10.27823313051667768280627807065, −9.047238683116572348753334660324, −8.194582223717480809532172846074, −7.53952198139795363691610221860, −6.65497367199788112935700236847, −5.97274465474525205495307987591, −5.51647721606033839988136203260, −3.88776663748944969612066883544, −3.24771223738240681988498975059, −2.50800642422116155029689283873, 1.02244064340270583080891420203, 1.99177430923159548210765465850, 3.56214251676088852567614910437, 4.19365788980586960072095405395, 5.27584181025116990752969768197, 5.69298688637660605581474900184, 6.63871768015587377744733379026, 8.255469278735420442249867143781, 8.917570597359165097244930879086, 9.142670813415578988894230288461

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