Properties

Label 2-418-11.3-c1-0-2
Degree $2$
Conductor $418$
Sign $-0.711 - 0.702i$
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 + (0.931 + 2.86i)3-s + (0.309 − 0.951i)4-s + (1.62 + 1.18i)5-s + (−2.43 − 1.77i)6-s + (0.750 − 2.30i)7-s + (0.309 + 0.951i)8-s + (−4.92 + 3.57i)9-s − 2.00·10-s + (2.41 + 2.27i)11-s + 3.01·12-s + (−2.45 + 1.78i)13-s + (0.750 + 2.30i)14-s + (−1.87 + 5.76i)15-s + (−0.809 − 0.587i)16-s + (3.49 + 2.53i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.537 + 1.65i)3-s + (0.154 − 0.475i)4-s + (0.727 + 0.528i)5-s + (−0.995 − 0.723i)6-s + (0.283 − 0.873i)7-s + (0.109 + 0.336i)8-s + (−1.64 + 1.19i)9-s − 0.635·10-s + (0.727 + 0.685i)11-s + 0.870·12-s + (−0.681 + 0.494i)13-s + (0.200 + 0.617i)14-s + (−0.483 + 1.48i)15-s + (−0.202 − 0.146i)16-s + (0.847 + 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.711 - 0.702i$
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: \((2,\ 418,\ (\ :1/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.529047 + 1.28782i\)
\(L(\frac12)\) \(\approx\) \(0.529047 + 1.28782i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 - 2.27i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.931 - 2.86i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.62 - 1.18i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.750 + 2.30i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.45 - 1.78i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.49 - 2.53i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 2.27T + 23T^{2} \)
29 \( 1 + (-2.72 + 8.38i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.51 - 5.46i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.330 + 1.01i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.59 + 4.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 + (2.78 + 8.55i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.12 - 2.99i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.48 - 4.55i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.49 + 5.44i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 9.19T + 67T^{2} \)
71 \( 1 + (-8.70 - 6.32i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.76 + 8.50i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.27 - 3.83i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.30 - 2.40i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.04T + 89T^{2} \)
97 \( 1 + (-7.62 + 5.54i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.93670633050114429893896628551, −10.33705269155194559025745442625, −9.740918344851895984296687879956, −9.165236364950167274022513434172, −8.013846277017493730283998214798, −7.00482935805464477289092583276, −5.80292611900343325499374849501, −4.61097422604292091561073836602, −3.76767178238231777700894083146, −2.17169788821605041270150613906, 1.11205448661914591887015545435, 2.11798947567755962061431752860, 3.17483007264351575336300689151, 5.39523554997789764265301721279, 6.24890707710193106970805290688, 7.43125258088061446808703814788, 8.131022535781763707664507688430, 9.035047215072549272871703560777, 9.535179015169623881953340924458, 11.07684534619639656886320530677

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