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Properties

Label 2-418-11.9-c1-0-1
Degree $2$
Conductor $418$
Sign $-0.353 - 0.935i$
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  + (−0.309 + 0.951i)2-s + (−0.971 + 0.705i)3-s + (−0.809 − 0.587i)4-s + (−1.28 − 3.96i)5-s + (−0.370 − 1.14i)6-s + (0.280 + 0.203i)7-s + (0.809 − 0.587i)8-s + (−0.481 + 1.48i)9-s + 4.16·10-s + (0.00601 + 3.31i)11-s + 1.20·12-s + (−0.839 + 2.58i)13-s + (−0.280 + 0.203i)14-s + (4.04 + 2.93i)15-s + (0.309 + 0.951i)16-s + (2.41 + 7.42i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.560 + 0.407i)3-s + (−0.404 − 0.293i)4-s + (−0.575 − 1.77i)5-s + (−0.151 − 0.466i)6-s + (0.105 + 0.0769i)7-s + (0.286 − 0.207i)8-s + (−0.160 + 0.494i)9-s + 1.31·10-s + (0.00181 + 0.999i)11-s + 0.346·12-s + (−0.232 + 0.716i)13-s + (−0.0748 + 0.0543i)14-s + (1.04 + 0.758i)15-s + (0.0772 + 0.237i)16-s + (0.585 + 1.80i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.353 - 0.935i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.404350 + 0.585327i\)
\(L(\frac12)\) \(\approx\) \(0.404350 + 0.585327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.00601 - 3.31i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (0.971 - 0.705i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (1.28 + 3.96i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.280 - 0.203i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.839 - 2.58i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.41 - 7.42i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 - 7.96T + 23T^{2} \)
29 \( 1 + (-6.18 - 4.49i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.437 + 1.34i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.95 + 3.59i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.40 - 3.92i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.21T + 43T^{2} \)
47 \( 1 + (-0.0345 + 0.0250i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.65 - 8.16i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.07 + 3.68i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.139 - 0.428i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 0.952T + 67T^{2} \)
71 \( 1 + (-4.38 - 13.5i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-11.4 - 8.31i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.95 - 12.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.85 + 5.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 3.81T + 89T^{2} \)
97 \( 1 + (-2.57 + 7.92i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.53542566769731778067621925973, −10.45909252239654430417684765720, −9.496593751221743150391933146609, −8.617732810966645502440519470648, −8.009372475366099034749780217591, −6.85420645926619196549673584689, −5.43846352989282534602895245597, −4.88988404935879070264951171176, −4.10986944978325184516381740089, −1.44289268216485773708310756082, 0.58175189990502636423866423409, 2.96213541907670154934216554656, 3.28778553469109446604211389735, 5.13102616429450472275841966735, 6.40731790397691726082720491944, 7.15722599804638259018120686165, 8.021641287856865360287878377078, 9.323272652761340473603765411981, 10.38053925933059830291908120822, 11.02500390845269679752198234540

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