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Properties

Label 2-418-209.151-c1-0-14
Degree $2$
Conductor $418$
Sign $0.273 + 0.961i$
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 + (1.03 − 0.336i)3-s + (0.309 − 0.951i)4-s + (−1.83 − 1.33i)5-s + (0.640 − 0.881i)6-s + (3.63 + 1.18i)7-s + (−0.309 − 0.951i)8-s + (−1.46 + 1.06i)9-s − 2.26·10-s + (−0.855 − 3.20i)11-s − 1.08i·12-s + (4.36 − 3.16i)13-s + (3.63 − 1.18i)14-s + (−2.34 − 0.762i)15-s + (−0.809 − 0.587i)16-s + (2.52 − 3.47i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.598 − 0.194i)3-s + (0.154 − 0.475i)4-s + (−0.819 − 0.595i)5-s + (0.261 − 0.359i)6-s + (1.37 + 0.447i)7-s + (−0.109 − 0.336i)8-s + (−0.488 + 0.355i)9-s − 0.715·10-s + (−0.258 − 0.966i)11-s − 0.314i·12-s + (1.20 − 0.878i)13-s + (0.972 − 0.316i)14-s + (−0.605 − 0.196i)15-s + (−0.202 − 0.146i)16-s + (0.611 − 0.842i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.273 + 0.961i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.273 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73582 - 1.31110i\)
\(L(\frac12)\) \(\approx\) \(1.73582 - 1.31110i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.855 + 3.20i)T \)
19 \( 1 + (-2.36 - 3.65i)T \)
good3 \( 1 + (-1.03 + 0.336i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.83 + 1.33i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-3.63 - 1.18i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.36 + 3.16i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.52 + 3.47i)T + (-5.25 - 16.1i)T^{2} \)
23 \( 1 + 8.61T + 23T^{2} \)
29 \( 1 + (2.29 - 7.06i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.19 - 1.65i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.16 - 0.703i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.39 - 4.29i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.82iT - 43T^{2} \)
47 \( 1 + (-0.991 - 3.05i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.91 - 2.63i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-7.56 - 2.45i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.0208 - 0.0286i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 10.6iT - 67T^{2} \)
71 \( 1 + (-2.67 + 3.67i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.84 + 1.24i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.84 + 7.14i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.50 - 3.44i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 0.748iT - 89T^{2} \)
97 \( 1 + (9.91 + 13.6i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.34615512589132744064874081916, −10.40054098686921359619693475212, −8.908019493951437449594943574361, −8.014752790954549944189338086210, −7.946913145200154412089770393079, −5.81385904596327307830096119541, −5.22257754886947685176244278417, −3.91347977600627146488670116701, −2.89490036302490215638126078663, −1.32316826663111774434208580099, 2.12715866828687535773129916659, 3.80460848787595856500100192952, 4.16088713468649590840895629386, 5.62147738214798049800426407170, 6.81544556128279817163607766829, 7.906397963073780745059543384817, 8.163224266616284721016944298542, 9.463778678320839042912842854568, 10.72603949779160927882596972137, 11.56709989160857301386072349863

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