Properties

Label 2-418-19.16-c1-0-12
Degree $2$
Conductor $418$
Sign $0.911 - 0.412i$
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  + (0.766 + 0.642i)2-s + (2.66 − 0.969i)3-s + (0.173 + 0.984i)4-s + (−0.115 + 0.652i)5-s + (2.66 + 0.969i)6-s + (0.738 + 1.27i)7-s + (−0.500 + 0.866i)8-s + (3.85 − 3.23i)9-s + (−0.507 + 0.425i)10-s + (0.5 − 0.866i)11-s + (1.41 + 2.45i)12-s + (−3.14 − 1.14i)13-s + (−0.256 + 1.45i)14-s + (0.326 + 1.84i)15-s + (−0.939 + 0.342i)16-s + (−3.62 − 3.04i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (1.53 − 0.559i)3-s + (0.0868 + 0.492i)4-s + (−0.0514 + 0.291i)5-s + (1.08 + 0.395i)6-s + (0.279 + 0.483i)7-s + (−0.176 + 0.306i)8-s + (1.28 − 1.07i)9-s + (−0.160 + 0.134i)10-s + (0.150 − 0.261i)11-s + (0.409 + 0.708i)12-s + (−0.873 − 0.317i)13-s + (−0.0685 + 0.388i)14-s + (0.0841 + 0.477i)15-s + (−0.234 + 0.0855i)16-s + (−0.879 − 0.738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.911 - 0.412i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.911 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.75994 + 0.595127i\)
\(L(\frac12)\) \(\approx\) \(2.75994 + 0.595127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (4.35 - 0.263i)T \)
good3 \( 1 + (-2.66 + 0.969i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (0.115 - 0.652i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.738 - 1.27i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (3.14 + 1.14i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.62 + 3.04i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.636 - 3.61i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.46 + 1.23i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.0850 + 0.147i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.52T + 37T^{2} \)
41 \( 1 + (-3.57 + 1.30i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.49 + 8.47i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-5.29 + 4.44i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.266 - 1.50i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (1.42 + 1.19i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.11 - 6.31i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (8.19 - 6.87i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.63 - 14.9i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (7.94 - 2.89i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (8.64 - 3.14i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (4.46 + 7.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.06 - 2.93i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (4.00 + 3.36i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.52257352703117224969412390548, −10.22962646632364429565418308570, −8.879522677090543900892924467715, −8.645560521129304856939836256144, −7.37141369155760453404778053947, −6.97773828268135138846638939136, −5.53250062104640731249219843959, −4.22304856436845460523154189466, −2.99610406272015107724724699478, −2.18312320509487924631852823893, 1.93566704442428071083297173203, 2.97404045023550251680090795180, 4.33732146405566163546752465960, 4.58690824672583115122272481647, 6.48125669943743307802358221032, 7.62124765267730588999052850766, 8.657051484102278594683883648079, 9.238988726602612116798533123128, 10.31746103880332109791235820652, 10.86993674890456952541458283813

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