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Properties

Label 2-603-67.37-c1-0-17
Degree $2$
Conductor $603$
Sign $-0.641 - 0.767i$
Analytic cond. $4.81497$
Root an. cond. $2.19430$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.31 − 2.28i)2-s + (−2.46 + 4.27i)4-s + 0.846·5-s + (0.241 − 0.417i)7-s + 7.73·8-s + (−1.11 − 1.92i)10-s + (−2.25 + 3.90i)11-s + (−0.355 − 0.615i)13-s − 1.26·14-s + (−5.24 − 9.08i)16-s + (−3.30 − 5.72i)17-s + (−2.96 − 5.12i)19-s + (−2.08 + 3.61i)20-s + 11.8·22-s + (−2.61 − 4.53i)23-s + ⋯
L(s)  = 1  + (−0.931 − 1.61i)2-s + (−1.23 + 2.13i)4-s + 0.378·5-s + (0.0911 − 0.157i)7-s + 2.73·8-s + (−0.352 − 0.610i)10-s + (−0.680 + 1.17i)11-s + (−0.0984 − 0.170i)13-s − 0.339·14-s + (−1.31 − 2.27i)16-s + (−0.802 − 1.38i)17-s + (−0.679 − 1.17i)19-s + (−0.466 + 0.808i)20-s + 2.53·22-s + (−0.545 − 0.944i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $-0.641 - 0.767i$
Analytic conductor: \(4.81497\)
Root analytic conductor: \(2.19430\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{603} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 603,\ (\ :1/2),\ -0.641 - 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.136989 + 0.293033i\)
\(L(\frac12)\) \(\approx\) \(0.136989 + 0.293033i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
67 \( 1 + (0.565 + 8.16i)T \)
good2 \( 1 + (1.31 + 2.28i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.846T + 5T^{2} \)
7 \( 1 + (-0.241 + 0.417i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.25 - 3.90i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.355 + 0.615i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.30 + 5.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.96 + 5.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.61 + 4.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.47 - 2.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.64 + 2.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.31 + 5.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.93 - 6.80i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 7.36T + 43T^{2} \)
47 \( 1 + (-4.62 + 8.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 + 7.46T + 59T^{2} \)
61 \( 1 + (-1.40 - 2.44i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (1.71 - 2.96i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.75 - 3.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.09 + 1.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.629 + 1.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 + (1.87 + 3.24i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.19316800689141539620892674200, −9.441853864768354706733826562245, −8.759226256699131691898711609556, −7.72484189783942117905360698495, −6.86777288812072714363248379791, −5.00577369023349208682781802705, −4.17685617921122203737111175497, −2.66457726749731307920275146704, −2.05470390366012449777690083828, −0.23476428128231705788506981243, 1.75991119961390388403687073943, 3.94141600833506152948019720157, 5.40709415127408539822577270713, 5.94018924758607434802013572262, 6.69823625716332548820234786518, 7.957717340212991118763651767255, 8.329286418610274259244436210053, 9.148018811421982582091113942685, 10.21369895947212313352921799064, 10.62063868562327168200823981796

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