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Properties

Label 2-418-209.208-c1-0-10
Degree $2$
Conductor $418$
Sign $0.382 + 0.924i$
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  − 2-s − 2.00i·3-s + 4-s + 1.36·5-s + 2.00i·6-s + 0.331i·7-s − 8-s − 1.03·9-s − 1.36·10-s + (3.31 + 0.0887i)11-s − 2.00i·12-s + 5.11·13-s − 0.331i·14-s − 2.74i·15-s + 16-s + 2.58i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15i·3-s + 0.5·4-s + 0.610·5-s + 0.819i·6-s + 0.125i·7-s − 0.353·8-s − 0.343·9-s − 0.431·10-s + (0.999 + 0.0267i)11-s − 0.579i·12-s + 1.41·13-s − 0.0887i·14-s − 0.707i·15-s + 0.250·16-s + 0.627i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.382 + 0.924i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.382 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01325 - 0.677547i\)
\(L(\frac12)\) \(\approx\) \(1.01325 - 0.677547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
11 \( 1 + (-3.31 - 0.0887i)T \)
19 \( 1 + (1.77 + 3.98i)T \)
good3 \( 1 + 2.00iT - 3T^{2} \)
5 \( 1 - 1.36T + 5T^{2} \)
7 \( 1 - 0.331iT - 7T^{2} \)
13 \( 1 - 5.11T + 13T^{2} \)
17 \( 1 - 2.58iT - 17T^{2} \)
23 \( 1 + 2.10T + 23T^{2} \)
29 \( 1 - 0.741T + 29T^{2} \)
31 \( 1 - 5.22iT - 31T^{2} \)
37 \( 1 + 6.14iT - 37T^{2} \)
41 \( 1 + 2.49T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + 4.21T + 47T^{2} \)
53 \( 1 - 7.44iT - 53T^{2} \)
59 \( 1 + 1.84iT - 59T^{2} \)
61 \( 1 + 2.61iT - 61T^{2} \)
67 \( 1 - 2.28iT - 67T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 - 9.08iT - 73T^{2} \)
79 \( 1 + 5.96T + 79T^{2} \)
83 \( 1 + 13.2iT - 83T^{2} \)
89 \( 1 - 16.4iT - 89T^{2} \)
97 \( 1 + 3.52iT - 97T^{2} \)
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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

−11.03203849874889703408451173575, −10.10413736750508160194473606039, −8.962461654353676979213555712694, −8.437007933853010805889241802787, −7.23203176374450171531211938199, −6.48974340560346695875494697425, −5.80157055739261055818689831721, −3.87504259152179511301721966887, −2.15653792522882343340518540329, −1.20113393568195663340293084588, 1.58282879084724722008541701956, 3.42112297213876411598424955529, 4.36279613019512073632240397851, 5.80796138618369741044932939098, 6.55055792329232303569825655431, 7.981247347919196315724428201403, 8.922423414204388520469902754780, 9.664964246643861167865189378227, 10.20598903602446338718567202220, 11.13324975489460499662533051508

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