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Properties

Label 2-1424-356.195-c0-0-0
Degree $2$
Conductor $1424$
Sign $0.957 + 0.289i$
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  + (1.37 − 1.19i)5-s + (0.909 + 0.415i)9-s + (−0.398 + 1.83i)13-s + (0.0801 + 0.273i)17-s + (0.328 − 2.28i)25-s + (−1.19 − 0.0855i)29-s + (−0.677 + 0.677i)37-s + (−0.415 − 1.90i)41-s + (1.74 − 0.512i)45-s + (−0.989 − 0.142i)49-s + (−1.03 − 1.61i)53-s + (−0.0855 + 0.114i)61-s + (1.63 + 2.99i)65-s + (0.698 + 1.53i)73-s + (0.654 + 0.755i)81-s + ⋯
L(s)  = 1  + (1.37 − 1.19i)5-s + (0.909 + 0.415i)9-s + (−0.398 + 1.83i)13-s + (0.0801 + 0.273i)17-s + (0.328 − 2.28i)25-s + (−1.19 − 0.0855i)29-s + (−0.677 + 0.677i)37-s + (−0.415 − 1.90i)41-s + (1.74 − 0.512i)45-s + (−0.989 − 0.142i)49-s + (−1.03 − 1.61i)53-s + (−0.0855 + 0.114i)61-s + (1.63 + 2.99i)65-s + (0.698 + 1.53i)73-s + (0.654 + 0.755i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1424\)    =    \(2^{4} \cdot 89\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(0.710668\)
Root analytic conductor: \(0.843011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1424} (1263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1424,\ (\ :0),\ 0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.419290240\)
\(L(\frac12)\) \(\approx\) \(1.419290240\)
\(L(1)\) 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 \( 1 + (-0.654 + 0.755i)T \)
good3 \( 1 + (-0.909 - 0.415i)T^{2} \)
5 \( 1 + (-1.37 + 1.19i)T + (0.142 - 0.989i)T^{2} \)
7 \( 1 + (0.989 + 0.142i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (0.398 - 1.83i)T + (-0.909 - 0.415i)T^{2} \)
17 \( 1 + (-0.0801 - 0.273i)T + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (0.755 - 0.654i)T^{2} \)
23 \( 1 + (-0.755 + 0.654i)T^{2} \)
29 \( 1 + (1.19 + 0.0855i)T + (0.989 + 0.142i)T^{2} \)
31 \( 1 + (-0.755 - 0.654i)T^{2} \)
37 \( 1 + (0.677 - 0.677i)T - iT^{2} \)
41 \( 1 + (0.415 + 1.90i)T + (-0.909 + 0.415i)T^{2} \)
43 \( 1 + (-0.989 + 0.142i)T^{2} \)
47 \( 1 + (0.415 + 0.909i)T^{2} \)
53 \( 1 + (1.03 + 1.61i)T + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (-0.909 + 0.415i)T^{2} \)
61 \( 1 + (0.0855 - 0.114i)T + (-0.281 - 0.959i)T^{2} \)
67 \( 1 + (-0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.654 + 0.755i)T^{2} \)
83 \( 1 + (-0.540 - 0.841i)T^{2} \)
97 \( 1 + (0.708 + 0.817i)T + (-0.142 + 0.989i)T^{2} \)
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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

−9.637566958126299672820580545516, −9.075044241781155329730275014627, −8.303019244730721339034543199303, −7.12563120918351807603465717172, −6.45207542002639489561804122779, −5.39606697678738345352533770100, −4.79034400019857020632588415121, −3.92679946372215215768235943326, −2.04565354004737773463141000692, −1.63578907028014317950389846568, 1.57259707603500095669653137496, 2.75405491924719339819169073726, 3.45490837855740757009898545199, 4.94130562273241535613496212256, 5.77588706154352625045594016013, 6.46215365980098722033431632229, 7.28478279494129624916442180335, 7.966767121955209788154123064555, 9.471028334810253855324755124885, 9.649303119105377896508410117433

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