Properties

Label 2-737-737.725-c0-0-0
Degree $2$
Conductor $737$
Sign $-0.0348 - 0.999i$
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  + (0.186 − 0.215i)3-s + (0.723 + 0.690i)4-s + (−1.67 + 1.07i)5-s + (0.130 + 0.909i)9-s + (−0.888 − 0.458i)11-s + (0.283 − 0.0270i)12-s + (−0.0806 + 0.560i)15-s + (0.0475 + 0.998i)16-s + (−1.95 − 0.376i)20-s + (0.428 + 1.23i)23-s + (1.23 − 2.69i)25-s + (0.459 + 0.295i)27-s + (−0.911 + 1.28i)31-s + (−0.264 + 0.105i)33-s + (−0.533 + 0.748i)36-s + (0.786 − 1.36i)37-s + ⋯
L(s)  = 1  + (0.186 − 0.215i)3-s + (0.723 + 0.690i)4-s + (−1.67 + 1.07i)5-s + (0.130 + 0.909i)9-s + (−0.888 − 0.458i)11-s + (0.283 − 0.0270i)12-s + (−0.0806 + 0.560i)15-s + (0.0475 + 0.998i)16-s + (−1.95 − 0.376i)20-s + (0.428 + 1.23i)23-s + (1.23 − 2.69i)25-s + (0.459 + 0.295i)27-s + (−0.911 + 1.28i)31-s + (−0.264 + 0.105i)33-s + (−0.533 + 0.748i)36-s + (0.786 − 1.36i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(737\)    =    \(11 \cdot 67\)
Sign: $-0.0348 - 0.999i$
Analytic conductor: \(0.367810\)
Root analytic conductor: \(0.606474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{737} (725, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 737,\ (\ :0),\ -0.0348 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8226225539\)
\(L(\frac12)\) \(\approx\) \(0.8226225539\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.888 + 0.458i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
good2 \( 1 + (-0.723 - 0.690i)T^{2} \)
3 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
5 \( 1 + (1.67 - 1.07i)T + (0.415 - 0.909i)T^{2} \)
7 \( 1 + (-0.235 + 0.971i)T^{2} \)
13 \( 1 + (0.327 - 0.945i)T^{2} \)
17 \( 1 + (-0.0475 + 0.998i)T^{2} \)
19 \( 1 + (-0.235 - 0.971i)T^{2} \)
23 \( 1 + (-0.428 - 1.23i)T + (-0.786 + 0.618i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.911 - 1.28i)T + (-0.327 - 0.945i)T^{2} \)
37 \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.888 - 0.458i)T^{2} \)
43 \( 1 + (-0.841 - 0.540i)T^{2} \)
47 \( 1 + (-1.82 - 0.351i)T + (0.928 + 0.371i)T^{2} \)
53 \( 1 + (-1.70 + 0.500i)T + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (0.738 + 1.61i)T + (-0.654 + 0.755i)T^{2} \)
61 \( 1 + (-0.580 + 0.814i)T^{2} \)
71 \( 1 + (-0.601 - 0.573i)T + (0.0475 + 0.998i)T^{2} \)
73 \( 1 + (-0.580 + 0.814i)T^{2} \)
79 \( 1 + (-0.981 + 0.189i)T^{2} \)
83 \( 1 + (0.995 - 0.0950i)T^{2} \)
89 \( 1 + (0.759 + 0.876i)T + (-0.142 + 0.989i)T^{2} \)
97 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.95073667370760698708871517328, −10.42415327798158492035987597726, −8.686151936316664394434540220497, −7.944107105061357402522520050884, −7.37641508520405151592267500679, −6.95336874162656302337800100761, −5.48562487770839808952568820819, −4.04530290148030246159685521429, −3.23748138422964453597294768692, −2.39538959573792573528500386974, 0.873462383892445419247479014050, 2.73619237564129383521428007081, 4.03685302839075371552736363398, 4.76475503930049359765615416367, 5.85397472077304493409472420872, 7.09838679343894132356778277216, 7.70824339224273907816777791634, 8.679404920443158836926645411643, 9.426823526051934919784054973874, 10.48333582175580054215138384680

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