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Properties

Label 2-1424-356.303-c0-0-0
Degree $2$
Conductor $1424$
Sign $0.910 - 0.413i$
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 + (1.86 − 0.697i)13-s + (1.03 + 1.61i)17-s + (1.23 + 0.361i)25-s + (0.767 − 0.574i)29-s + (1.38 + 1.38i)37-s + (0.654 + 0.244i)41-s + (1.27 − 0.817i)45-s + (−0.281 + 0.959i)49-s + (−1.53 − 0.698i)53-s + (0.574 − 1.05i)61-s + (−2.94 + 0.640i)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 + (1.86 − 0.697i)13-s + (1.03 + 1.61i)17-s + (1.23 + 0.361i)25-s + (0.767 − 0.574i)29-s + (1.38 + 1.38i)37-s + (0.654 + 0.244i)41-s + (1.27 − 0.817i)45-s + (−0.281 + 0.959i)49-s + (−1.53 − 0.698i)53-s + (0.574 − 1.05i)61-s + (−2.94 + 0.640i)65-s + (−0.544 + 0.627i)73-s + (0.142 − 0.989i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1424\)    =    \(2^{4} \cdot 89\)
Sign: $0.910 - 0.413i$
Analytic conductor: \(0.710668\)
Root analytic conductor: \(0.843011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1424} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1424,\ (\ :0),\ 0.910 - 0.413i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8523889543\)
\(L(\frac12)\) \(\approx\) \(0.8523889543\)
\(L(1)\) 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 \( 1 + (-1.86 + 0.697i)T + (0.755 - 0.654i)T^{2} \)
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 \( 1 + (-0.767 + 0.574i)T + (0.281 - 0.959i)T^{2} \)
31 \( 1 + (-0.989 + 0.142i)T^{2} \)
37 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
41 \( 1 + (-0.654 - 0.244i)T + (0.755 + 0.654i)T^{2} \)
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 \( 1 + (-0.574 + 1.05i)T + (-0.540 - 0.841i)T^{2} \)
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

−9.862563826041081780125543305883, −8.560830485698988795675257257699, −8.081827826111020913891385140883, −7.892384561599014960829749146769, −6.38127025623217096767190300802, −5.77144998269390710964839271675, −4.59381563990090201346282566150, −3.73346551641756486213087756378, −3.03198202548960908776186556391, −1.20384752121580218936488131932, 0.901078262296299197271732907269, 2.92365798865522940791511431224, 3.60853606499037534013605668157, 4.39363290273518208291188815560, 5.61659179633860092466777739533, 6.49121353970116918434893611903, 7.34370217860636456562047126790, 8.076589142672310305950791192241, 8.835816093607437541849424535452, 9.457733887830317042690728673866

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