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Properties

Label 2-603-67.37-c1-0-13
Degree $2$
Conductor $603$
Sign $-0.986 + 0.164i$
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  + (−1.04 − 1.81i)2-s + (−1.20 + 2.08i)4-s + 1.25·5-s + (−1.51 + 2.62i)7-s + 0.846·8-s + (−1.31 − 2.27i)10-s + (2.09 − 3.63i)11-s + (−0.798 − 1.38i)13-s + 6.35·14-s + (1.51 + 2.62i)16-s + (−0.423 − 0.733i)17-s + (−3.21 − 5.57i)19-s + (−1.50 + 2.60i)20-s − 8.80·22-s + (−2.97 − 5.15i)23-s + ⋯
L(s)  = 1  + (−0.741 − 1.28i)2-s + (−0.600 + 1.04i)4-s + 0.559·5-s + (−0.572 + 0.991i)7-s + 0.299·8-s + (−0.415 − 0.719i)10-s + (0.632 − 1.09i)11-s + (−0.221 − 0.383i)13-s + 1.69·14-s + (0.378 + 0.656i)16-s + (−0.102 − 0.177i)17-s + (−0.737 − 1.27i)19-s + (−0.336 + 0.582i)20-s − 1.87·22-s + (−0.620 − 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $-0.986 + 0.164i$
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: \((2,\ 603,\ (\ :1/2),\ -0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0609603 - 0.735783i\)
\(L(\frac12)\) \(\approx\) \(0.0609603 - 0.735783i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.32 - 3.64i)T \)
good2 \( 1 + (1.04 + 1.81i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 1.25T + 5T^{2} \)
7 \( 1 + (1.51 - 2.62i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.09 + 3.63i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.798 + 1.38i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.423 + 0.733i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.21 + 5.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.97 + 5.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.60 + 6.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.02 + 6.98i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.23 - 2.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.55 + 4.42i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 0.596T + 43T^{2} \)
47 \( 1 + (1.30 - 2.25i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.54T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + (-3.51 - 6.08i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (2.97 - 5.15i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.92 - 8.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.60 + 7.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.67 + 2.90i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.89T + 89T^{2} \)
97 \( 1 + (-3.88 - 6.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.14739699948859990893084178730, −9.501914632090063266155050227027, −8.831016745302131353106877544341, −8.129639762375996451219129617920, −6.35422752437493839573240061394, −5.90982575515785394424075508641, −4.25622114043201521227289619921, −2.86705912139640748307074260738, −2.28741722121265802377971174890, −0.52277350290506057209769105706, 1.61647535436002280490917945953, 3.60907750208138105320011028157, 4.81699309607958142840544595075, 6.11701003923202504618335018547, 6.65993806281248268943412104577, 7.44132744304630967365274255275, 8.266748315884856109313218240574, 9.425100182222334999147647751612, 9.812105357571740899457744650919, 10.58689730867755363208526445473

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