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Properties

Label 2-603-9.4-c1-0-34
Degree $2$
Conductor $603$
Sign $-0.656 + 0.753i$
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  + (−1.38 − 2.40i)2-s + (1.42 − 0.980i)3-s + (−2.86 + 4.95i)4-s + (1.47 − 2.54i)5-s + (−4.34 − 2.07i)6-s + (2.08 + 3.60i)7-s + 10.3·8-s + (1.07 − 2.79i)9-s − 8.18·10-s + (1.31 + 2.27i)11-s + (0.771 + 9.88i)12-s + (−0.699 + 1.21i)13-s + (5.79 − 10.0i)14-s + (−0.397 − 5.08i)15-s + (−8.64 − 14.9i)16-s + 1.96·17-s + ⋯
L(s)  = 1  + (−0.982 − 1.70i)2-s + (0.824 − 0.565i)3-s + (−1.43 + 2.47i)4-s + (0.658 − 1.14i)5-s + (−1.77 − 0.846i)6-s + (0.787 + 1.36i)7-s + 3.65·8-s + (0.359 − 0.933i)9-s − 2.58·10-s + (0.396 + 0.686i)11-s + (0.222 + 2.85i)12-s + (−0.194 + 0.336i)13-s + (1.54 − 2.68i)14-s + (−0.102 − 1.31i)15-s + (−2.16 − 3.74i)16-s + 0.476·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $-0.656 + 0.753i$
Analytic conductor: \(4.81497\)
Root analytic conductor: \(2.19430\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{603} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 603,\ (\ :1/2),\ -0.656 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.579818 - 1.27421i\)
\(L(\frac12)\) \(\approx\) \(0.579818 - 1.27421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.42 + 0.980i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.38 + 2.40i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.47 + 2.54i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.08 - 3.60i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.31 - 2.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.699 - 1.21i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.96T + 17T^{2} \)
19 \( 1 - 7.27T + 19T^{2} \)
23 \( 1 + (0.710 - 1.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.86 + 6.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.585 - 1.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.804T + 37T^{2} \)
41 \( 1 + (1.45 - 2.51i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.32 - 4.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.44 + 9.42i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.65T + 53T^{2} \)
59 \( 1 + (6.16 - 10.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.275 + 0.477i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + 3.15T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + (-0.423 - 0.733i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.86 - 6.69i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.702T + 89T^{2} \)
97 \( 1 + (9.12 + 15.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.869717197381955071981097508364, −9.466941728057299429383988108962, −8.910494911225650451167330256063, −8.186960665417295143347381152821, −7.43095754139526427989925297353, −5.43340958935467650973215037197, −4.34825571702966484011256555065, −2.97144361257717932585115249070, −1.91473560296544275950951646116, −1.35930726384035831414217847599, 1.39864906241687573627049936890, 3.46684875961827436142415479130, 4.80427600879804616611329162523, 5.72251386495704458757150109228, 6.90459484300558379884605339429, 7.49790379697094053062725054384, 8.083649425847728710982342313520, 9.201333574150347204198136884754, 9.836345235493334399444090066589, 10.59174202282131058112848562607

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