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Properties

Label 2-1424-356.347-c0-0-0
Degree $2$
Conductor $1424$
Sign $-0.840 - 0.542i$
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 + (−1.59 − 0.114i)13-s + (1.53 + 0.698i)17-s + (−2.45 − 1.57i)25-s + (−1.71 + 0.936i)29-s + (−0.494 + 0.494i)37-s + (0.142 − 0.0101i)41-s + (0.822 − 1.80i)45-s + (−0.540 + 0.841i)49-s + (0.627 − 0.544i)53-s + (0.936 + 0.203i)61-s + (1.10 − 2.96i)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 + (−1.59 − 0.114i)13-s + (1.53 + 0.698i)17-s + (−2.45 − 1.57i)25-s + (−1.71 + 0.936i)29-s + (−0.494 + 0.494i)37-s + (0.142 − 0.0101i)41-s + (0.822 − 1.80i)45-s + (−0.540 + 0.841i)49-s + (0.627 − 0.544i)53-s + (0.936 + 0.203i)61-s + (1.10 − 2.96i)65-s + (0.186 + 1.29i)73-s + (0.959 + 0.281i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1424\)    =    \(2^{4} \cdot 89\)
Sign: $-0.840 - 0.542i$
Analytic conductor: \(0.710668\)
Root analytic conductor: \(0.843011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1424} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1424,\ (\ :0),\ -0.840 - 0.542i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5832690900\)
\(L(\frac12)\) \(\approx\) \(0.5832690900\)
\(L(1)\) 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 \( 1 + (1.59 + 0.114i)T + (0.989 + 0.142i)T^{2} \)
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 \( 1 + (1.71 - 0.936i)T + (0.540 - 0.841i)T^{2} \)
31 \( 1 + (-0.281 - 0.959i)T^{2} \)
37 \( 1 + (0.494 - 0.494i)T - iT^{2} \)
41 \( 1 + (-0.142 + 0.0101i)T + (0.989 - 0.142i)T^{2} \)
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 \( 1 + (-0.936 - 0.203i)T + (0.909 + 0.415i)T^{2} \)
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

−10.18316887815648047458100463372, −9.468310671135310897303855164241, −8.205562744565079044218208378602, −7.54360621992492271575866764087, −6.98566713097804349066805365102, −6.02469424579730325827400306955, −5.24657327656149818037210563058, −3.74106487271421317829113908749, −3.14668522292361701166839877990, −2.27264280822312128486484318306, 0.45066103869204078958311151694, 2.02543488495123454388327031890, 3.41024441038203912324678062051, 4.47627556337014661189548151657, 5.30699283121272758909271107832, 5.64701325072294175836054868938, 7.34871557708148683422800785289, 7.83940491707335096659008563502, 8.598929751311829728560464131564, 9.427864511511951046879419506199

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