Properties

Label 2-418-19.5-c1-0-5
Degree $2$
Conductor $418$
Sign $-0.197 - 0.980i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
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  + (−0.939 + 0.342i)2-s + (−0.235 + 1.33i)3-s + (0.766 − 0.642i)4-s + (2.14 + 1.80i)5-s + (−0.235 − 1.33i)6-s + (2.28 + 3.96i)7-s + (−0.500 + 0.866i)8-s + (1.08 + 0.395i)9-s + (−2.63 − 0.958i)10-s + (0.5 − 0.866i)11-s + (0.678 + 1.17i)12-s + (−1.08 − 6.13i)13-s + (−3.50 − 2.93i)14-s + (−2.91 + 2.44i)15-s + (0.173 − 0.984i)16-s + (2.83 − 1.03i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−0.136 + 0.771i)3-s + (0.383 − 0.321i)4-s + (0.959 + 0.805i)5-s + (−0.0962 − 0.545i)6-s + (0.864 + 1.49i)7-s + (−0.176 + 0.306i)8-s + (0.362 + 0.131i)9-s + (−0.832 − 0.303i)10-s + (0.150 − 0.261i)11-s + (0.195 + 0.339i)12-s + (−0.300 − 1.70i)13-s + (−0.936 − 0.785i)14-s + (−0.752 + 0.631i)15-s + (0.0434 − 0.246i)16-s + (0.687 − 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.197 - 0.980i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.197 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816904 + 0.997458i\)
\(L(\frac12)\) \(\approx\) \(0.816904 + 0.997458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (2.85 - 3.29i)T \)
good3 \( 1 + (0.235 - 1.33i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-2.14 - 1.80i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-2.28 - 3.96i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (1.08 + 6.13i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-2.83 + 1.03i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-5.86 + 4.91i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.89 + 1.78i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.70 - 2.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.26T + 37T^{2} \)
41 \( 1 + (0.0499 - 0.283i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (6.74 + 5.65i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (9.82 + 3.57i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-2.65 + 2.22i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-11.1 + 4.07i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (4.80 - 4.03i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-3.04 - 1.10i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-8.00 - 6.71i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-2.21 + 12.5i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.0785 + 0.445i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (3.07 + 5.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.72 + 9.78i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (11.2 - 4.08i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.05207481374051048991532451589, −10.35432453235193285883001742044, −9.848884862230014828467558132823, −8.762505053624446670236777300378, −8.029171718595363643963274774142, −6.73127967413265696068690156926, −5.56746132333414267960440200728, −5.17112614249369337610738254299, −3.11393014196007881454941467979, −1.96280798562854926490586793383, 1.25521553641948788512630565751, 1.78513535057421083860735225917, 4.05046224410107378030673373148, 5.07676870128287306981438657942, 6.70064766198031329270401477448, 7.14821828558142950405811334489, 8.166039767979652101431504174133, 9.335704858864067476997922987692, 9.825506049390933303050115000970, 11.01780738850078464220570263655

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