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Properties

Label 2-418-11.3-c1-0-15
Degree $2$
Conductor $418$
Sign $-0.998 - 0.0450i$
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 + (−0.494 − 1.52i)3-s + (0.309 − 0.951i)4-s + (−3.03 − 2.20i)5-s + (−1.29 − 0.939i)6-s + (−0.259 + 0.797i)7-s + (−0.309 − 0.951i)8-s + (0.358 − 0.260i)9-s − 3.74·10-s + (3.31 + 0.0707i)11-s − 1.59·12-s + (−3.59 + 2.61i)13-s + (0.259 + 0.797i)14-s + (−1.85 + 5.69i)15-s + (−0.809 − 0.587i)16-s + (−4.06 − 2.95i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.285 − 0.877i)3-s + (0.154 − 0.475i)4-s + (−1.35 − 0.985i)5-s + (−0.528 − 0.383i)6-s + (−0.0979 + 0.301i)7-s + (−0.109 − 0.336i)8-s + (0.119 − 0.0868i)9-s − 1.18·10-s + (0.999 + 0.0213i)11-s − 0.461·12-s + (−0.996 + 0.724i)13-s + (0.0692 + 0.213i)14-s + (−0.478 + 1.47i)15-s + (−0.202 − 0.146i)16-s + (−0.986 − 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.998 - 0.0450i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.998 - 0.0450i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0236427 + 1.04834i\)
\(L(\frac12)\) \(\approx\) \(0.0236427 + 1.04834i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 - 0.0707i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.494 + 1.52i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (3.03 + 2.20i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.259 - 0.797i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.59 - 2.61i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.06 + 2.95i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 6.07T + 23T^{2} \)
29 \( 1 + (-0.862 + 2.65i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.99 + 4.35i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.52 + 7.77i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.97 - 6.07i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.55T + 43T^{2} \)
47 \( 1 + (2.52 + 7.77i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.8 + 7.85i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.25 + 6.93i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.68 + 4.85i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.32T + 67T^{2} \)
71 \( 1 + (10.0 + 7.29i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.33 - 7.18i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.26 - 5.28i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.05 - 0.768i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + (6.56 - 4.77i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.44368588074638824158885624375, −9.757627958538414262509457725365, −8.960578741120291366488856899547, −7.78986436861286264244634024922, −6.99340870016461616172792425377, −5.98777717458776372244917902685, −4.50889016516274006115400672084, −4.04791403614063953697390122027, −2.15176540744749943920949493746, −0.58084908189244947081046703754, 2.92089932889007177937849030607, 4.12158609408426340185385543707, 4.43838476449547007395265045838, 6.00248100294705782123569486601, 7.05530490607434264992143650358, 7.67434198842750451773916615460, 8.836308245397196325495436521906, 10.25593517102031992269091245259, 10.68221701585552374701961123206, 11.74197030763562116841853248440

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