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Properties

Label 2-3344-209.75-c0-0-1
Degree $2$
Conductor $3344$
Sign $0.772 + 0.634i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
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  + (−0.0646 − 0.198i)5-s + (1.47 + 1.07i)7-s + (0.309 − 0.951i)9-s + (−0.913 − 0.406i)11-s + (−0.604 − 1.86i)17-s + (0.809 − 0.587i)19-s − 0.618·23-s + (0.773 − 0.562i)25-s + (0.118 − 0.363i)35-s − 1.33·43-s − 0.209·45-s + (1.08 − 0.786i)47-s + (0.722 + 2.22i)49-s + (−0.0218 + 0.207i)55-s + (0.413 + 1.27i)61-s + ⋯
L(s)  = 1  + (−0.0646 − 0.198i)5-s + (1.47 + 1.07i)7-s + (0.309 − 0.951i)9-s + (−0.913 − 0.406i)11-s + (−0.604 − 1.86i)17-s + (0.809 − 0.587i)19-s − 0.618·23-s + (0.773 − 0.562i)25-s + (0.118 − 0.363i)35-s − 1.33·43-s − 0.209·45-s + (1.08 − 0.786i)47-s + (0.722 + 2.22i)49-s + (−0.0218 + 0.207i)55-s + (0.413 + 1.27i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.772 + 0.634i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (1329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ 0.772 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.412204391\)
\(L(\frac12)\) \(\approx\) \(1.412204391\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.0646 + 0.198i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-1.47 - 1.07i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.604 + 1.86i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + 0.618T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 1.33T + T^{2} \)
47 \( 1 + (-1.08 + 0.786i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−8.767519036661018329128693023669, −8.069634747853557740493888601220, −7.30973313839621953694747299463, −6.52724586426050709582865293083, −5.31716492993744211749463033568, −5.17320729446569768862363954395, −4.22500093902858812970833306934, −2.93060497224455802261274714088, −2.28714738806929781821899086723, −0.914355804684205294424322723752, 1.49079358451271958918693356124, 2.08974183355695739063359943758, 3.50965478291085018301487789022, 4.39456384997138807985853641633, 4.92309293100906155805269360818, 5.73153541186229024582972784641, 6.86083119018638253012385434748, 7.61508588824473221097305938245, 7.990920837611746732055160228107, 8.554535234583504697233712620705

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