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Properties

Label 2-1424-356.131-c0-0-0
Degree $2$
Conductor $1424$
Sign $-0.0206 - 0.999i$
Analytic cond. $0.710668$
Root an. cond. $0.843011$
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  + (−1.49 + 0.215i)5-s + (0.755 + 0.654i)9-s + (0.0498 − 0.133i)13-s + (−1.03 + 1.61i)17-s + (1.23 − 0.361i)25-s + (−1.05 + 1.40i)29-s + (0.300 + 0.300i)37-s + (0.654 + 1.75i)41-s + (−1.27 − 0.817i)45-s + (0.281 + 0.959i)49-s + (1.53 − 0.698i)53-s + (−1.40 + 0.767i)61-s + (−0.0458 + 0.210i)65-s + (−0.544 − 0.627i)73-s + (0.142 + 0.989i)81-s + ⋯
L(s)  = 1  + (−1.49 + 0.215i)5-s + (0.755 + 0.654i)9-s + (0.0498 − 0.133i)13-s + (−1.03 + 1.61i)17-s + (1.23 − 0.361i)25-s + (−1.05 + 1.40i)29-s + (0.300 + 0.300i)37-s + (0.654 + 1.75i)41-s + (−1.27 − 0.817i)45-s + (0.281 + 0.959i)49-s + (1.53 − 0.698i)53-s + (−1.40 + 0.767i)61-s + (−0.0458 + 0.210i)65-s + (−0.544 − 0.627i)73-s + (0.142 + 0.989i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1424\)    =    \(2^{4} \cdot 89\)
Sign: $-0.0206 - 0.999i$
Analytic conductor: \(0.710668\)
Root analytic conductor: \(0.843011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1424} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1424,\ (\ :0),\ -0.0206 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6908950181\)
\(L(\frac12)\) \(\approx\) \(0.6908950181\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 + (-0.142 + 0.989i)T \)
good3 \( 1 + (-0.755 - 0.654i)T^{2} \)
5 \( 1 + (1.49 - 0.215i)T + (0.959 - 0.281i)T^{2} \)
7 \( 1 + (-0.281 - 0.959i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (-0.0498 + 0.133i)T + (-0.755 - 0.654i)T^{2} \)
17 \( 1 + (1.03 - 1.61i)T + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.989 + 0.142i)T^{2} \)
23 \( 1 + (0.989 - 0.142i)T^{2} \)
29 \( 1 + (1.05 - 1.40i)T + (-0.281 - 0.959i)T^{2} \)
31 \( 1 + (0.989 + 0.142i)T^{2} \)
37 \( 1 + (-0.300 - 0.300i)T + iT^{2} \)
41 \( 1 + (-0.654 - 1.75i)T + (-0.755 + 0.654i)T^{2} \)
43 \( 1 + (0.281 - 0.959i)T^{2} \)
47 \( 1 + (-0.654 - 0.755i)T^{2} \)
53 \( 1 + (-1.53 + 0.698i)T + (0.654 - 0.755i)T^{2} \)
59 \( 1 + (-0.755 + 0.654i)T^{2} \)
61 \( 1 + (1.40 - 0.767i)T + (0.540 - 0.841i)T^{2} \)
67 \( 1 + (0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
79 \( 1 + (-0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.909 + 0.415i)T^{2} \)
97 \( 1 + (0.258 + 1.80i)T + (-0.959 + 0.281i)T^{2} \)
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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

−10.10860092667315537835808512612, −8.940109752504525278500719612297, −8.236014867859257152866761378467, −7.55377080545404138179569813135, −6.92394381727672017631817094388, −5.88363672831284221770243379129, −4.57594541851334834685735514848, −4.11342115810844456281937827692, −3.10538443225973502443100912899, −1.65185901491239214099501731411, 0.58643323335842644552160719415, 2.41585713002446961384654983641, 3.78815673265911605944893223409, 4.19989755863485579727296931843, 5.19251463615621447538831100381, 6.46237982523685797215416724450, 7.33266864265883322184000287051, 7.67786992394814475597154044135, 8.888146700247714618274694697059, 9.290647034487193865707894798203

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