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Properties

Label 2-418-11.4-c1-0-9
Degree $2$
Conductor $418$
Sign $0.776 - 0.630i$
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 + (0.430 − 1.32i)3-s + (0.309 + 0.951i)4-s + (−1.74 + 1.27i)5-s + (1.12 − 0.819i)6-s + (0.365 + 1.12i)7-s + (−0.309 + 0.951i)8-s + (0.853 + 0.620i)9-s − 2.16·10-s + (2.62 − 2.02i)11-s + 1.39·12-s + (5.04 + 3.66i)13-s + (−0.365 + 1.12i)14-s + (0.931 + 2.86i)15-s + (−0.809 + 0.587i)16-s + (1.63 − 1.19i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.248 − 0.765i)3-s + (0.154 + 0.475i)4-s + (−0.781 + 0.567i)5-s + (0.460 − 0.334i)6-s + (0.138 + 0.424i)7-s + (−0.109 + 0.336i)8-s + (0.284 + 0.206i)9-s − 0.683·10-s + (0.790 − 0.611i)11-s + 0.402·12-s + (1.40 + 1.01i)13-s + (−0.0976 + 0.300i)14-s + (0.240 + 0.739i)15-s + (−0.202 + 0.146i)16-s + (0.397 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.776 - 0.630i$
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: \((2,\ 418,\ (\ :1/2),\ 0.776 - 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86284 + 0.661213i\)
\(L(\frac12)\) \(\approx\) \(1.86284 + 0.661213i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 + 2.02i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.430 + 1.32i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.74 - 1.27i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.365 - 1.12i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-5.04 - 3.66i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.63 + 1.19i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 + (1.07 + 3.29i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.52 + 4.74i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.55 - 4.78i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.739 + 2.27i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.95T + 43T^{2} \)
47 \( 1 + (-0.939 + 2.89i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.912 - 0.662i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.756 + 2.32i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (9.11 - 6.61i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + (0.469 - 0.341i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.55 + 7.85i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.48 - 3.98i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-14.3 + 10.4i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 5.72T + 89T^{2} \)
97 \( 1 + (-2.07 - 1.50i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.71766836723637518428625155066, −10.71289768578234742591229556594, −9.182693558565773848113577556888, −8.264671219753836492326254004394, −7.51051190792068426939882281824, −6.61725009312932409311654712579, −5.86268742954147043963294123401, −4.23166284763138128761643230466, −3.41840072774511876780155370483, −1.79874418428324746228063398502, 1.28861635675080247262177759140, 3.50007290353519976659745007183, 3.96843326817305941459681509323, 4.91544244364513084911511443378, 6.18541270565457209786569569081, 7.44482324443378655557277270323, 8.529450603044020553466828563353, 9.380815514059190110754431988999, 10.41636626882605943375152415380, 10.99205961831807457356210559300

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