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Properties

Label 2-418-11.4-c1-0-3
Degree $2$
Conductor $418$
Sign $0.879 - 0.475i$
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 + (0.636 − 1.95i)3-s + (0.309 + 0.951i)4-s + (−1.58 + 1.15i)5-s + (−1.66 + 1.21i)6-s + (1.27 + 3.92i)7-s + (0.309 − 0.951i)8-s + (−1.00 − 0.729i)9-s + 1.96·10-s + (−2.95 + 1.50i)11-s + 2.05·12-s + (3.85 + 2.80i)13-s + (1.27 − 3.92i)14-s + (1.25 + 3.84i)15-s + (−0.809 + 0.587i)16-s + (−5.67 + 4.12i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.367 − 1.13i)3-s + (0.154 + 0.475i)4-s + (−0.710 + 0.516i)5-s + (−0.680 + 0.494i)6-s + (0.481 + 1.48i)7-s + (0.109 − 0.336i)8-s + (−0.334 − 0.243i)9-s + 0.621·10-s + (−0.891 + 0.453i)11-s + 0.594·12-s + (1.07 + 0.777i)13-s + (0.340 − 1.04i)14-s + (0.322 + 0.993i)15-s + (−0.202 + 0.146i)16-s + (−1.37 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.879 - 0.475i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.914509 + 0.231075i\)
\(L(\frac12)\) \(\approx\) \(0.914509 + 0.231075i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.95 - 1.50i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.636 + 1.95i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.58 - 1.15i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.27 - 3.92i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.85 - 2.80i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.67 - 4.12i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 - 3.10T + 23T^{2} \)
29 \( 1 + (0.208 + 0.642i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.656 - 0.476i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.63 + 11.1i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.55 + 7.87i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.01T + 43T^{2} \)
47 \( 1 + (1.86 - 5.74i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.98 - 2.89i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.88 - 5.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.99 + 2.17i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 + (7.01 - 5.09i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.07 - 9.47i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-11.6 - 8.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.01 + 1.46i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.51T + 89T^{2} \)
97 \( 1 + (11.5 + 8.37i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.20077436384762923946691390522, −10.72455822002780247531652372537, −9.034120000535735220014188368111, −8.590971423371618842440356889076, −7.69585939496427120238267687730, −6.92582554292307967444253271684, −5.80254345186853600018294643095, −4.11792520037568859385595225873, −2.56097818532462507185250978702, −1.83558399847013565309598681299, 0.72065981447254705377619457310, 3.23053504860274100236974288734, 4.38958440007955067170706664617, 4.98885504620637028424747622525, 6.64194317301158738589725655599, 7.74105156175546321637075161453, 8.397782284031214071005275338801, 9.198025037185490726137899700282, 10.37820553475318825761381525953, 10.73817553900415837722832576367

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