Properties

Label 2-418-11.3-c1-0-8
Degree $2$
Conductor $418$
Sign $0.0694 + 0.997i$
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 + (−0.618 − 1.90i)3-s + (0.309 − 0.951i)4-s + (0.5 + 0.363i)5-s + (1.61 + 1.17i)6-s + (0.572 − 1.76i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 0.618·10-s + (0.309 + 3.30i)11-s − 2·12-s + (5.23 − 3.80i)13-s + (0.572 + 1.76i)14-s + (0.381 − 1.17i)15-s + (−0.809 − 0.587i)16-s + (−4.54 − 3.30i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.356 − 1.09i)3-s + (0.154 − 0.475i)4-s + (0.223 + 0.162i)5-s + (0.660 + 0.479i)6-s + (0.216 − 0.666i)7-s + (0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s − 0.195·10-s + (0.0931 + 0.995i)11-s − 0.577·12-s + (1.45 − 1.05i)13-s + (0.153 + 0.471i)14-s + (0.0986 − 0.303i)15-s + (−0.202 − 0.146i)16-s + (−1.10 − 0.800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.0694 + 0.997i$
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: \((2,\ 418,\ (\ :1/2),\ 0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692142 - 0.645614i\)
\(L(\frac12)\) \(\approx\) \(0.692142 - 0.645614i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 3.30i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.618 + 1.90i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.5 - 0.363i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.572 + 1.76i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-5.23 + 3.80i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.54 + 3.30i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 1.14T + 23T^{2} \)
29 \( 1 + (-0.472 + 1.45i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.61 - 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.23 + 3.80i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.85 + 8.78i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.09T + 43T^{2} \)
47 \( 1 + (-1.11 - 3.44i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.47 + 5.42i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.85 - 11.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.54 + 6.20i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 7.23T + 67T^{2} \)
71 \( 1 + (-10.4 - 7.60i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.32 - 13.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.85 + 5.70i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.73 + 6.34i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 + (-3.85 + 2.80i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.86736934125407575588815597761, −10.18129680825917826210515318658, −9.030696353601354222213351095659, −7.977599680916252134459120038520, −7.23487653529941524357698586948, −6.53063217404418710586333437155, −5.62085977439024149242964255414, −4.12668965013348972973821882914, −2.14963870579464030505250677736, −0.816497378725594445056197074180, 1.74210905738127679859352162734, 3.47535636387407131184010383135, 4.40856939543952486267972775354, 5.67165545256966001338407811505, 6.56221347873411026291673372828, 8.228207805233759227710467716902, 8.944067473281747064412794803406, 9.495556106536690218621830331644, 10.75108662041694652053733828899, 11.10250748931018417562092341363

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