Properties

Label 2-418-19.17-c1-0-1
Degree $2$
Conductor $418$
Sign $-0.600 + 0.799i$
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  + (−0.173 + 0.984i)2-s + (1.31 + 1.10i)3-s + (−0.939 − 0.342i)4-s + (−3.37 + 1.22i)5-s + (−1.31 + 1.10i)6-s + (−1.75 − 3.03i)7-s + (0.5 − 0.866i)8-s + (−0.00647 − 0.0367i)9-s + (−0.623 − 3.53i)10-s + (−0.5 + 0.866i)11-s + (−0.860 − 1.49i)12-s + (−4.44 + 3.72i)13-s + (3.29 − 1.19i)14-s + (−5.81 − 2.11i)15-s + (0.766 + 0.642i)16-s + (0.138 − 0.787i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (0.761 + 0.638i)3-s + (−0.469 − 0.171i)4-s + (−1.51 + 0.549i)5-s + (−0.538 + 0.451i)6-s + (−0.661 − 1.14i)7-s + (0.176 − 0.306i)8-s + (−0.00215 − 0.0122i)9-s + (−0.197 − 1.11i)10-s + (−0.150 + 0.261i)11-s + (−0.248 − 0.430i)12-s + (−1.23 + 1.03i)13-s + (0.879 − 0.320i)14-s + (−1.50 − 0.546i)15-s + (0.191 + 0.160i)16-s + (0.0336 − 0.191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.600 + 0.799i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0852649 - 0.170643i\)
\(L(\frac12)\) \(\approx\) \(0.0852649 - 0.170643i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (4.33 + 0.455i)T \)
good3 \( 1 + (-1.31 - 1.10i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (3.37 - 1.22i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.75 + 3.03i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (4.44 - 3.72i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.138 + 0.787i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (2.27 + 0.828i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.21 - 6.89i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-0.937 - 1.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.30T + 37T^{2} \)
41 \( 1 + (-7.35 - 6.16i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (7.07 - 2.57i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.742 + 4.21i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-5.86 - 2.13i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.222 + 1.26i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (12.1 + 4.42i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.71 - 15.4i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (0.213 - 0.0778i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (7.09 + 5.95i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (0.701 + 0.588i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (5.06 + 8.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (12.5 - 10.5i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-1.92 + 10.9i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.77106764458811239720696284318, −10.58816499879630212564094082656, −9.923846981712474920342173380900, −8.966658936577721667432008674644, −8.032911115082583600563760061056, −7.12094009858072468210773885122, −6.68824971796728521762015676857, −4.56601241810951774810236077452, −4.05706205388872019177227006193, −3.03642460181063566092227612906, 0.11061470685213029877188875853, 2.30575133636661107553887825479, 3.17399342769802961399921228270, 4.42571706206009692083317877184, 5.68279139904656175520507212778, 7.31830685329106521160465078910, 8.148646008545692044190241192665, 8.542142380970995871993292127292, 9.594558659789087169374872146857, 10.70893994018891542065584436144

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