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Properties

Label 2-418-19.11-c1-0-7
Degree $2$
Conductor $418$
Sign $0.910 + 0.412i$
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  + (−0.5 + 0.866i)2-s + (1 − 1.73i)3-s + (−0.499 − 0.866i)4-s + (0.999 + 1.73i)6-s + 2·7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + 11-s − 1.99·12-s + (0.5 + 0.866i)13-s + (−1 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (3 − 5.19i)17-s + 0.999·18-s + (−3.5 − 2.59i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 − 0.433i)4-s + (0.408 + 0.707i)6-s + 0.755·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + 0.301·11-s − 0.577·12-s + (0.138 + 0.240i)13-s + (−0.267 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (0.727 − 1.26i)17-s + 0.235·18-s + (−0.802 − 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46406 - 0.316424i\)
\(L(\frac12)\) \(\approx\) \(1.46406 - 0.316424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 - T \)
19 \( 1 + (3.5 + 2.59i)T \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.5 - 7.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.22682234217004666619911428810, −10.04888732771626787755320277732, −8.976529768976431548031177304582, −8.285603398092119200046154165400, −7.40561654795960395919066679000, −6.84542093230910216464929138403, −5.56151146100428010617863353430, −4.40904864180515439500091636305, −2.59351915792078184431253759349, −1.24521425287284537056541760796, 1.67531807233687475215544391971, 3.27876004991778180731223911297, 4.08903152934446920283427507627, 5.13831236350976290999180483901, 6.62770548705313872563619376033, 8.234048812911613774936302853707, 8.467779840250957616275657126665, 9.600714985020813527948544248553, 10.40060935443841962456662611276, 10.88001311492037414070644940726

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