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Properties

Label 2-418-209.5-c1-0-1
Degree $2$
Conductor $418$
Sign $-0.745 - 0.666i$
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  + (−0.615 − 0.788i)2-s + (−0.215 + 0.208i)3-s + (−0.241 + 0.970i)4-s + (−0.160 + 0.100i)5-s + (0.296 + 0.0417i)6-s + (−1.81 − 0.806i)7-s + (0.913 − 0.406i)8-s + (−0.101 + 2.90i)9-s + (0.177 + 0.0647i)10-s + (−2.25 − 2.43i)11-s + (−0.149 − 0.259i)12-s + (−2.74 + 1.45i)13-s + (0.479 + 1.92i)14-s + (0.0137 − 0.0550i)15-s + (−0.882 − 0.469i)16-s + (−0.0244 − 0.700i)17-s + ⋯
L(s)  = 1  + (−0.435 − 0.557i)2-s + (−0.124 + 0.120i)3-s + (−0.120 + 0.485i)4-s + (−0.0717 + 0.0448i)5-s + (0.121 + 0.0170i)6-s + (−0.684 − 0.304i)7-s + (0.322 − 0.143i)8-s + (−0.0338 + 0.969i)9-s + (0.0562 + 0.0204i)10-s + (−0.680 − 0.733i)11-s + (−0.0432 − 0.0749i)12-s + (−0.760 + 0.404i)13-s + (0.128 + 0.514i)14-s + (0.00354 − 0.0142i)15-s + (−0.220 − 0.117i)16-s + (−0.00593 − 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.745 - 0.666i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.745 - 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0513654 + 0.134408i\)
\(L(\frac12)\) \(\approx\) \(0.0513654 + 0.134408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.615 + 0.788i)T \)
11 \( 1 + (2.25 + 2.43i)T \)
19 \( 1 + (4.35 + 0.174i)T \)
good3 \( 1 + (0.215 - 0.208i)T + (0.104 - 2.99i)T^{2} \)
5 \( 1 + (0.160 - 0.100i)T + (2.19 - 4.49i)T^{2} \)
7 \( 1 + (1.81 + 0.806i)T + (4.68 + 5.20i)T^{2} \)
13 \( 1 + (2.74 - 1.45i)T + (7.26 - 10.7i)T^{2} \)
17 \( 1 + (0.0244 + 0.700i)T + (-16.9 + 1.18i)T^{2} \)
23 \( 1 + (-0.196 + 0.164i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (8.18 - 2.34i)T + (24.5 - 15.3i)T^{2} \)
31 \( 1 + (-0.876 + 0.973i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (1.02 - 0.748i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.49 - 1.44i)T + (1.43 - 40.9i)T^{2} \)
43 \( 1 + (0.430 + 0.360i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-4.83 - 7.16i)T + (-17.6 + 43.5i)T^{2} \)
53 \( 1 + (-6.32 - 3.95i)T + (23.2 + 47.6i)T^{2} \)
59 \( 1 + (2.40 - 3.56i)T + (-22.1 - 54.7i)T^{2} \)
61 \( 1 + (1.78 + 4.40i)T + (-43.8 + 42.3i)T^{2} \)
67 \( 1 + (7.50 + 2.73i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-11.9 + 7.49i)T + (31.1 - 63.8i)T^{2} \)
73 \( 1 + (-0.166 - 0.341i)T + (-44.9 + 57.5i)T^{2} \)
79 \( 1 + (3.75 - 0.528i)T + (75.9 - 21.7i)T^{2} \)
83 \( 1 + (-13.5 + 2.87i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 + (1.30 + 7.42i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (2.30 + 2.94i)T + (-23.4 + 94.1i)T^{2} \)
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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

−11.22643541031251525365352181966, −10.73410452809692476137246003560, −9.856009268563546327076031159994, −8.991595685232091550095919596254, −7.911061496694206204260539172694, −7.14937353569473551228295227593, −5.79208905799302110530195602018, −4.62679430292734537777106700929, −3.35313392721849659373667782201, −2.13300914475235970260957513107, 0.099757532288837620779754962356, 2.33469929503468902803583084241, 3.93589777489592212505299630452, 5.29870379005097630861594754594, 6.25313535414873468155455275283, 7.09129232513449768017089503673, 8.036590816545093065303306295067, 9.088034133396947339549365356969, 9.834797056694531983698211255752, 10.56376256808201598889252143120

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