Properties

Label 2-1424-356.107-c0-0-0
Degree $2$
Conductor $1424$
Sign $0.992 + 0.124i$
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  + (−0.304 − 0.474i)5-s + (0.281 + 0.959i)9-s + (0.574 − 0.767i)13-s + (0.627 + 0.544i)17-s + (0.283 − 0.620i)25-s + (1.83 + 0.398i)29-s + (−1.41 − 1.41i)37-s + (0.959 + 1.28i)41-s + (0.368 − 0.425i)45-s + (−0.909 − 0.415i)49-s + (1.29 − 0.186i)53-s + (−0.398 − 0.148i)61-s + (−0.538 − 0.0385i)65-s + (0.273 + 0.0801i)73-s + (−0.841 + 0.540i)81-s + ⋯
L(s)  = 1  + (−0.304 − 0.474i)5-s + (0.281 + 0.959i)9-s + (0.574 − 0.767i)13-s + (0.627 + 0.544i)17-s + (0.283 − 0.620i)25-s + (1.83 + 0.398i)29-s + (−1.41 − 1.41i)37-s + (0.959 + 1.28i)41-s + (0.368 − 0.425i)45-s + (−0.909 − 0.415i)49-s + (1.29 − 0.186i)53-s + (−0.398 − 0.148i)61-s + (−0.538 − 0.0385i)65-s + (0.273 + 0.0801i)73-s + (−0.841 + 0.540i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1424\)    =    \(2^{4} \cdot 89\)
Sign: $0.992 + 0.124i$
Analytic conductor: \(0.710668\)
Root analytic conductor: \(0.843011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1424} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1424,\ (\ :0),\ 0.992 + 0.124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.109085700\)
\(L(\frac12)\) \(\approx\) \(1.109085700\)
\(L(1)\) 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 \( 1 + (0.841 + 0.540i)T \)
good3 \( 1 + (-0.281 - 0.959i)T^{2} \)
5 \( 1 + (0.304 + 0.474i)T + (-0.415 + 0.909i)T^{2} \)
7 \( 1 + (0.909 + 0.415i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (-0.574 + 0.767i)T + (-0.281 - 0.959i)T^{2} \)
17 \( 1 + (-0.627 - 0.544i)T + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (-0.540 - 0.841i)T^{2} \)
23 \( 1 + (0.540 + 0.841i)T^{2} \)
29 \( 1 + (-1.83 - 0.398i)T + (0.909 + 0.415i)T^{2} \)
31 \( 1 + (0.540 - 0.841i)T^{2} \)
37 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
41 \( 1 + (-0.959 - 1.28i)T + (-0.281 + 0.959i)T^{2} \)
43 \( 1 + (-0.909 + 0.415i)T^{2} \)
47 \( 1 + (-0.959 - 0.281i)T^{2} \)
53 \( 1 + (-1.29 + 0.186i)T + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.281 + 0.959i)T^{2} \)
61 \( 1 + (0.398 + 0.148i)T + (0.755 + 0.654i)T^{2} \)
67 \( 1 + (0.959 - 0.281i)T^{2} \)
71 \( 1 + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.989 - 0.142i)T^{2} \)
97 \( 1 + (1.66 - 1.07i)T + (0.415 - 0.909i)T^{2} \)
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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

−9.864842722631609553095531069910, −8.637304327174503604530468264479, −8.242396666668869828646447181878, −7.46026499519195032209488610995, −6.43755670070663672985683005590, −5.46452175794068922153004859450, −4.73146462612577929723291386068, −3.76404051691202706251974806923, −2.62342257719319214647635010841, −1.23310033476632181331293654967, 1.26821178756102249703269645693, 2.84923202894841296994521971791, 3.67599151260641255117820138670, 4.58387912385787175925784208679, 5.72443884798063038877140853571, 6.69816850960380746116020946638, 7.09463471076677640643764825485, 8.219744521017655625909167358417, 8.968653970849591312574582136299, 9.749167156205042607447885274850

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