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Properties

Label 2-737-737.571-c0-0-0
Degree $2$
Conductor $737$
Sign $-0.971 - 0.236i$
Analytic cond. $0.367810$
Root an. cond. $0.606474$
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  + (0.273 + 1.89i)3-s + (−0.888 − 0.458i)4-s + (−0.271 + 0.595i)5-s + (−2.57 + 0.755i)9-s + (−0.995 + 0.0950i)11-s + (0.627 − 1.81i)12-s + (−1.20 − 0.353i)15-s + (0.580 + 0.814i)16-s + (0.514 − 0.404i)20-s + (−0.264 − 0.105i)23-s + (0.374 + 0.432i)25-s + (−1.34 − 2.93i)27-s + (1.42 + 0.273i)31-s + (−0.452 − 1.86i)33-s + (2.63 + 0.507i)36-s + (−0.723 + 1.25i)37-s + ⋯
L(s)  = 1  + (0.273 + 1.89i)3-s + (−0.888 − 0.458i)4-s + (−0.271 + 0.595i)5-s + (−2.57 + 0.755i)9-s + (−0.995 + 0.0950i)11-s + (0.627 − 1.81i)12-s + (−1.20 − 0.353i)15-s + (0.580 + 0.814i)16-s + (0.514 − 0.404i)20-s + (−0.264 − 0.105i)23-s + (0.374 + 0.432i)25-s + (−1.34 − 2.93i)27-s + (1.42 + 0.273i)31-s + (−0.452 − 1.86i)33-s + (2.63 + 0.507i)36-s + (−0.723 + 1.25i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(737\)    =    \(11 \cdot 67\)
Sign: $-0.971 - 0.236i$
Analytic conductor: \(0.367810\)
Root analytic conductor: \(0.606474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{737} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 737,\ (\ :0),\ -0.971 - 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6069281213\)
\(L(\frac12)\) \(\approx\) \(0.6069281213\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (-0.928 + 0.371i)T \)
good2 \( 1 + (0.888 + 0.458i)T^{2} \)
3 \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \)
5 \( 1 + (0.271 - 0.595i)T + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.0475 - 0.998i)T^{2} \)
13 \( 1 + (-0.928 + 0.371i)T^{2} \)
17 \( 1 + (-0.580 + 0.814i)T^{2} \)
19 \( 1 + (-0.0475 + 0.998i)T^{2} \)
23 \( 1 + (0.264 + 0.105i)T + (0.723 + 0.690i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1.42 - 0.273i)T + (0.928 + 0.371i)T^{2} \)
37 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.995 + 0.0950i)T^{2} \)
43 \( 1 + (-0.415 - 0.909i)T^{2} \)
47 \( 1 + (0.370 - 0.291i)T + (0.235 - 0.971i)T^{2} \)
53 \( 1 + (1.67 - 1.07i)T + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (-1.30 + 1.50i)T + (-0.142 - 0.989i)T^{2} \)
61 \( 1 + (-0.981 - 0.189i)T^{2} \)
71 \( 1 + (-1.16 - 0.600i)T + (0.580 + 0.814i)T^{2} \)
73 \( 1 + (-0.981 - 0.189i)T^{2} \)
79 \( 1 + (0.786 + 0.618i)T^{2} \)
83 \( 1 + (0.327 - 0.945i)T^{2} \)
89 \( 1 + (0.279 - 1.94i)T + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.72939702782407210496494352517, −10.01473204922227460330628084238, −9.579721012519320878291022514358, −8.563250581683822228265082270335, −7.981410962688335866415814480323, −6.30680291203111815521450685959, −5.16609910291080863634452033425, −4.72355944442024107502830023252, −3.67768641965168193184073725535, −2.82390744651427554007395587825, 0.63608141169059796707986662926, 2.26080977152802948409920279808, 3.40172539331668539823056400032, 4.85768295179950800322367240887, 5.80190457505834798012033394032, 6.92998431524217817426142758718, 7.78536159822136676133039614219, 8.327873673817914781675140151782, 8.839904421622808212433839315819, 10.05628898354393303982773847336

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