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Properties

Label 2-603-67.25-c1-0-26
Degree $2$
Conductor $603$
Sign $-0.994 + 0.100i$
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  + (−0.220 − 1.53i)2-s + (−0.374 + 0.110i)4-s + (0.601 − 0.386i)5-s + (−0.391 − 2.72i)7-s + (−1.03 − 2.26i)8-s + (−0.724 − 0.835i)10-s + (−3.38 + 2.17i)11-s + (0.379 − 0.832i)13-s + (−4.07 + 1.19i)14-s + (−3.89 + 2.50i)16-s + (0.709 + 0.208i)17-s + (0.962 − 6.69i)19-s + (−0.183 + 0.211i)20-s + (4.07 + 4.70i)22-s + (1.79 − 2.07i)23-s + ⋯
L(s)  = 1  + (−0.155 − 1.08i)2-s + (−0.187 + 0.0550i)4-s + (0.269 − 0.172i)5-s + (−0.147 − 1.02i)7-s + (−0.365 − 0.800i)8-s + (−0.229 − 0.264i)10-s + (−1.02 + 0.656i)11-s + (0.105 − 0.230i)13-s + (−1.09 + 0.320i)14-s + (−0.973 + 0.625i)16-s + (0.172 + 0.0505i)17-s + (0.220 − 1.53i)19-s + (−0.0409 + 0.0472i)20-s + (0.869 + 1.00i)22-s + (0.375 − 0.432i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(4.81497\)
Root analytic conductor: \(2.19430\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{603} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 603,\ (\ :1/2),\ -0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0600091 - 1.19303i\)
\(L(\frac12)\) \(\approx\) \(0.0600091 - 1.19303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.96 + 4.29i)T \)
good2 \( 1 + (0.220 + 1.53i)T + (-1.91 + 0.563i)T^{2} \)
5 \( 1 + (-0.601 + 0.386i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (0.391 + 2.72i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (3.38 - 2.17i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-0.379 + 0.832i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.709 - 0.208i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-0.962 + 6.69i)T + (-18.2 - 5.35i)T^{2} \)
23 \( 1 + (-1.79 + 2.07i)T + (-3.27 - 22.7i)T^{2} \)
29 \( 1 - 2.93T + 29T^{2} \)
31 \( 1 + (1.93 + 4.23i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + 9.21T + 37T^{2} \)
41 \( 1 + (5.27 + 1.54i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-8.65 - 2.53i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (-7.37 + 8.51i)T + (-6.68 - 46.5i)T^{2} \)
53 \( 1 + (7.29 - 2.14i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-4.81 - 10.5i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-1.10 - 0.710i)T + (25.3 + 55.4i)T^{2} \)
71 \( 1 + (0.936 - 0.274i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-2.98 - 1.91i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (4.62 - 10.1i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-12.0 + 7.73i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (4.71 + 5.43i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 - 2.34T + 97T^{2} \)
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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

−10.42372458869459383797441896709, −9.681870627471398476342283415621, −8.800111624895323332597694134336, −7.42443418639915900259785995663, −6.87757301303828140272058108620, −5.45928389738478093588059487964, −4.35551979823910185776932484537, −3.20684973109054256193427264571, −2.14098014581204963466086970247, −0.66795626669810595608214244917, 2.18751565949888116367777673191, 3.30565607537856393568988092299, 5.15264187155510398108723120653, 5.76274095986799167378422190157, 6.47656749900861703365043927621, 7.60391948011730691498338264060, 8.313550280289896945838539197869, 9.014697925794990118229502570278, 10.11489936604421230272656295241, 11.00279029084851153980425041338

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