Properties

Label 2-418-209.151-c1-0-19
Degree $2$
Conductor $418$
Sign $-0.494 + 0.868i$
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 + (2.05 − 0.667i)3-s + (0.309 − 0.951i)4-s + (−2.88 − 2.09i)5-s + (1.26 − 1.74i)6-s + (−3.81 − 1.24i)7-s + (−0.309 − 0.951i)8-s + (1.34 − 0.977i)9-s − 3.56·10-s + (3.13 + 1.09i)11-s − 2.15i·12-s + (2.29 − 1.66i)13-s + (−3.81 + 1.24i)14-s + (−7.32 − 2.38i)15-s + (−0.809 − 0.587i)16-s + (−0.276 + 0.379i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (1.18 − 0.385i)3-s + (0.154 − 0.475i)4-s + (−1.29 − 0.937i)5-s + (0.518 − 0.713i)6-s + (−1.44 − 0.468i)7-s + (−0.109 − 0.336i)8-s + (0.448 − 0.325i)9-s − 1.12·10-s + (0.943 + 0.330i)11-s − 0.623i·12-s + (0.636 − 0.462i)13-s + (−1.02 + 0.331i)14-s + (−1.89 − 0.614i)15-s + (−0.202 − 0.146i)16-s + (−0.0669 + 0.0921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.494 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.494 + 0.868i$
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: \((2,\ 418,\ (\ :1/2),\ -0.494 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980142 - 1.68615i\)
\(L(\frac12)\) \(\approx\) \(0.980142 - 1.68615i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.13 - 1.09i)T \)
19 \( 1 + (-4.05 + 1.60i)T \)
good3 \( 1 + (-2.05 + 0.667i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.88 + 2.09i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (3.81 + 1.24i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.29 + 1.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.276 - 0.379i)T + (-5.25 - 16.1i)T^{2} \)
23 \( 1 - 3.48T + 23T^{2} \)
29 \( 1 + (-2.40 + 7.40i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.40 - 1.92i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-4.27 - 1.38i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.29 - 7.05i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.09iT - 43T^{2} \)
47 \( 1 + (-2.71 - 8.34i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.13 + 9.81i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (10.2 + 3.33i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.24 + 1.71i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.0989iT - 67T^{2} \)
71 \( 1 + (9.18 - 12.6i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.92 + 0.624i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.69 + 4.13i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.32 + 3.20i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 3.76iT - 89T^{2} \)
97 \( 1 + (-4.41 - 6.08i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.17706859969651083138315777833, −9.746584799001634644475794595299, −9.130867467693740232749159053165, −8.160168199052380912767932728127, −7.30979195789620447080807534799, −6.25664595445587559945558391966, −4.59606691348273192357118888544, −3.65950085706718679471182599402, −3.00412795200200921347828561836, −0.991569040593865135774732019349, 2.99921774301721859012905826543, 3.35750232665904241258469999516, 4.18100144409935809491809378617, 6.05439958717977162487950314892, 6.87077638704430574157300751432, 7.71268911980864758164708549092, 8.852963771522985548869697462887, 9.322682653567226645775472597038, 10.66096161715556614641361973665, 11.68148418316947525131488044675

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