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Properties

Label 2-418-209.18-c1-0-16
Degree $2$
Conductor $418$
Sign $0.497 - 0.867i$
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  + (0.809 + 0.587i)2-s + (2.71 + 0.881i)3-s + (0.309 + 0.951i)4-s + (−1.18 + 0.859i)5-s + (1.67 + 2.30i)6-s + (3.23 − 1.04i)7-s + (−0.309 + 0.951i)8-s + (4.15 + 3.02i)9-s − 1.46·10-s + (−1.63 − 2.88i)11-s + 2.85i·12-s + (−3.93 − 2.85i)13-s + (3.23 + 1.04i)14-s + (−3.96 + 1.28i)15-s + (−0.809 + 0.587i)16-s + (−1.99 − 2.74i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (1.56 + 0.509i)3-s + (0.154 + 0.475i)4-s + (−0.529 + 0.384i)5-s + (0.684 + 0.942i)6-s + (1.22 − 0.396i)7-s + (−0.109 + 0.336i)8-s + (1.38 + 1.00i)9-s − 0.462·10-s + (−0.494 − 0.869i)11-s + 0.823i·12-s + (−1.09 − 0.792i)13-s + (0.863 + 0.280i)14-s + (−1.02 + 0.333i)15-s + (−0.202 + 0.146i)16-s + (−0.484 − 0.666i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.497 - 0.867i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.497 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47579 + 1.43509i\)
\(L(\frac12)\) \(\approx\) \(2.47579 + 1.43509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (1.63 + 2.88i)T \)
19 \( 1 + (4.35 + 0.109i)T \)
good3 \( 1 + (-2.71 - 0.881i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.18 - 0.859i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-3.23 + 1.04i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.93 + 2.85i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.99 + 2.74i)T + (-5.25 + 16.1i)T^{2} \)
23 \( 1 + 1.03T + 23T^{2} \)
29 \( 1 + (-2.76 - 8.52i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.42 - 1.95i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-11.3 + 3.68i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.50 - 4.62i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 12.3iT - 43T^{2} \)
47 \( 1 + (1.19 - 3.68i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.42 + 1.95i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (6.62 - 2.15i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.50 + 4.82i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.43iT - 67T^{2} \)
71 \( 1 + (-5.58 - 7.68i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-11.7 + 3.80i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.96 - 4.33i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.294 - 0.405i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 5.68iT - 89T^{2} \)
97 \( 1 + (-0.135 + 0.185i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.15350458684839317756762826029, −10.59278444555647926341279180391, −9.302290500440772755875515169995, −8.282513741551293862268048248077, −7.86474581104297635507050351915, −7.04010691444133641195589248540, −5.24444571003532313286858733692, −4.38847634439418552677204746846, −3.33801809713124767464762107137, −2.40752745640345916664627229501, 1.94385440994279688308539183382, 2.46190286167297894846800474725, 4.24134400023777681490235413174, 4.63740462864344136879150048047, 6.43937265571475216413129250425, 7.81005760521791114615013670682, 8.049501302532689350996409633304, 9.153117731121734378339390504348, 10.01457238954900541105160900150, 11.32439382450669136259434080405

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