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Properties

Label 2-1424-356.227-c0-0-0
Degree $2$
Conductor $1424$
Sign $0.955 + 0.294i$
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  + (0.983 − 0.449i)5-s + (0.540 − 0.841i)9-s + (0.936 + 1.71i)13-s + (−1.29 − 0.186i)17-s + (0.110 − 0.127i)25-s + (0.697 − 1.86i)29-s + (−1.13 − 1.13i)37-s + (−0.841 + 1.54i)41-s + (0.153 − 1.07i)45-s + (0.755 + 0.654i)49-s + (0.0801 − 0.273i)53-s + (1.86 + 0.133i)61-s + (1.69 + 1.26i)65-s + (−1.61 + 1.03i)73-s + (−0.415 − 0.909i)81-s + ⋯
L(s)  = 1  + (0.983 − 0.449i)5-s + (0.540 − 0.841i)9-s + (0.936 + 1.71i)13-s + (−1.29 − 0.186i)17-s + (0.110 − 0.127i)25-s + (0.697 − 1.86i)29-s + (−1.13 − 1.13i)37-s + (−0.841 + 1.54i)41-s + (0.153 − 1.07i)45-s + (0.755 + 0.654i)49-s + (0.0801 − 0.273i)53-s + (1.86 + 0.133i)61-s + (1.69 + 1.26i)65-s + (−1.61 + 1.03i)73-s + (−0.415 − 0.909i)81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1424\)    =    \(2^{4} \cdot 89\)
Sign: $0.955 + 0.294i$
Analytic conductor: \(0.710668\)
Root analytic conductor: \(0.843011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1424} (1295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1424,\ (\ :0),\ 0.955 + 0.294i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.339072047\)
\(L(\frac12)\) \(\approx\) \(1.339072047\)
\(L(1)\) 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 \( 1 + (0.415 - 0.909i)T \)
good3 \( 1 + (-0.540 + 0.841i)T^{2} \)
5 \( 1 + (-0.983 + 0.449i)T + (0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.755 - 0.654i)T^{2} \)
11 \( 1 + (0.654 + 0.755i)T^{2} \)
13 \( 1 + (-0.936 - 1.71i)T + (-0.540 + 0.841i)T^{2} \)
17 \( 1 + (1.29 + 0.186i)T + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.909 - 0.415i)T^{2} \)
23 \( 1 + (-0.909 + 0.415i)T^{2} \)
29 \( 1 + (-0.697 + 1.86i)T + (-0.755 - 0.654i)T^{2} \)
31 \( 1 + (-0.909 - 0.415i)T^{2} \)
37 \( 1 + (1.13 + 1.13i)T + iT^{2} \)
41 \( 1 + (0.841 - 1.54i)T + (-0.540 - 0.841i)T^{2} \)
43 \( 1 + (0.755 - 0.654i)T^{2} \)
47 \( 1 + (0.841 - 0.540i)T^{2} \)
53 \( 1 + (-0.0801 + 0.273i)T + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.540 - 0.841i)T^{2} \)
61 \( 1 + (-1.86 - 0.133i)T + (0.989 + 0.142i)T^{2} \)
67 \( 1 + (-0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.415 - 0.909i)T^{2} \)
83 \( 1 + (0.281 - 0.959i)T^{2} \)
97 \( 1 + (0.234 + 0.512i)T + (-0.654 + 0.755i)T^{2} \)
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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

−9.516020593319303931678760684249, −9.076437478069340614434177858346, −8.358736091920947632152696760692, −6.94492383953610682244642579089, −6.50099096937696951494048250948, −5.70903672222616597650940416972, −4.48959370198257533413505686479, −3.91815721147512889506378801341, −2.33581342492558681827798932771, −1.40303794844723111444434395656, 1.56969593650424681616139502582, 2.61125561348821110848260720301, 3.64485771395382599998765927512, 4.96538556683273134096297843005, 5.58636019098298374029543763023, 6.55315792514004337777198748338, 7.20854235043872729873175204154, 8.363591499494375388785084665238, 8.808219006112282144012765876916, 10.11435167255807473363335027676

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