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Properties

Label 2-1424-356.187-c0-0-0
Degree $2$
Conductor $1424$
Sign $-0.294 - 0.955i$
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  + (−0.557 + 1.89i)5-s + (0.989 + 0.142i)9-s + (−0.0855 + 1.19i)13-s + (−1.53 − 0.698i)17-s + (−2.45 − 1.57i)25-s + (−0.203 − 0.373i)29-s + (1.32 + 1.32i)37-s + (0.142 + 1.98i)41-s + (−0.822 + 1.80i)45-s + (0.540 − 0.841i)49-s + (−0.627 + 0.544i)53-s + (0.373 − 1.71i)61-s + (−2.22 − 0.829i)65-s + (0.186 + 1.29i)73-s + (0.959 + 0.281i)81-s + ⋯
L(s)  = 1  + (−0.557 + 1.89i)5-s + (0.989 + 0.142i)9-s + (−0.0855 + 1.19i)13-s + (−1.53 − 0.698i)17-s + (−2.45 − 1.57i)25-s + (−0.203 − 0.373i)29-s + (1.32 + 1.32i)37-s + (0.142 + 1.98i)41-s + (−0.822 + 1.80i)45-s + (0.540 − 0.841i)49-s + (−0.627 + 0.544i)53-s + (0.373 − 1.71i)61-s + (−2.22 − 0.829i)65-s + (0.186 + 1.29i)73-s + (0.959 + 0.281i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1424\)    =    \(2^{4} \cdot 89\)
Sign: $-0.294 - 0.955i$
Analytic conductor: \(0.710668\)
Root analytic conductor: \(0.843011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1424} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1424,\ (\ :0),\ -0.294 - 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9219438438\)
\(L(\frac12)\) \(\approx\) \(0.9219438438\)
\(L(1)\) 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 \( 1 + (-0.959 + 0.281i)T \)
good3 \( 1 + (-0.989 - 0.142i)T^{2} \)
5 \( 1 + (0.557 - 1.89i)T + (-0.841 - 0.540i)T^{2} \)
7 \( 1 + (-0.540 + 0.841i)T^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (0.0855 - 1.19i)T + (-0.989 - 0.142i)T^{2} \)
17 \( 1 + (1.53 + 0.698i)T + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (-0.281 + 0.959i)T^{2} \)
23 \( 1 + (0.281 - 0.959i)T^{2} \)
29 \( 1 + (0.203 + 0.373i)T + (-0.540 + 0.841i)T^{2} \)
31 \( 1 + (0.281 + 0.959i)T^{2} \)
37 \( 1 + (-1.32 - 1.32i)T + iT^{2} \)
41 \( 1 + (-0.142 - 1.98i)T + (-0.989 + 0.142i)T^{2} \)
43 \( 1 + (0.540 + 0.841i)T^{2} \)
47 \( 1 + (-0.142 - 0.989i)T^{2} \)
53 \( 1 + (0.627 - 0.544i)T + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.989 + 0.142i)T^{2} \)
61 \( 1 + (-0.373 + 1.71i)T + (-0.909 - 0.415i)T^{2} \)
67 \( 1 + (0.142 - 0.989i)T^{2} \)
71 \( 1 + (0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (0.755 - 0.654i)T^{2} \)
97 \( 1 + (-1.45 - 0.425i)T + (0.841 + 0.540i)T^{2} \)
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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

−9.932392227134851586926984867079, −9.456089052629457816843881366803, −8.150017651269538216666153305253, −7.39774616340424375436743012243, −6.64315386398272344911496348620, −6.41745349459090386291772401352, −4.64904490437896497927963227248, −4.04852954301256583487008771676, −2.91858747427859007641240339210, −2.04531562083395684968245079198, 0.77186340835395204256367859383, 2.02567917064709953675982831039, 3.80533206629050899441713337910, 4.37381088333255359264084100398, 5.19438839638784378092200819843, 6.04515090406855847942038654275, 7.34730843715203912609801244726, 7.911453793821637059260489672682, 8.867041001971732863775014900633, 9.171881126709512439053944050750

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