Properties

Label 2-418-209.18-c1-0-12
Degree $2$
Conductor $418$
Sign $0.581 - 0.813i$
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 + (2.42 + 0.787i)3-s + (0.309 + 0.951i)4-s + (1.61 − 1.17i)5-s + (1.49 + 2.06i)6-s + (−2.37 + 0.770i)7-s + (−0.309 + 0.951i)8-s + (2.82 + 2.05i)9-s + 1.99·10-s + (−2.62 + 2.02i)11-s + 2.54i·12-s + (0.306 + 0.222i)13-s + (−2.37 − 0.770i)14-s + (4.83 − 1.57i)15-s + (−0.809 + 0.587i)16-s + (−1.98 − 2.72i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (1.39 + 0.454i)3-s + (0.154 + 0.475i)4-s + (0.722 − 0.525i)5-s + (0.611 + 0.841i)6-s + (−0.896 + 0.291i)7-s + (−0.109 + 0.336i)8-s + (0.941 + 0.684i)9-s + 0.631·10-s + (−0.792 + 0.610i)11-s + 0.735i·12-s + (0.0851 + 0.0618i)13-s + (−0.634 − 0.206i)14-s + (1.24 − 0.406i)15-s + (−0.202 + 0.146i)16-s + (−0.480 − 0.661i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50428 + 1.28742i\)
\(L(\frac12)\) \(\approx\) \(2.50428 + 1.28742i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.14 + 4.20i)T \)
good3 \( 1 + (-2.42 - 0.787i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.61 + 1.17i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (2.37 - 0.770i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.306 - 0.222i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.98 + 2.72i)T + (-5.25 + 16.1i)T^{2} \)
23 \( 1 - 9.33T + 23T^{2} \)
29 \( 1 + (2.55 + 7.87i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.17 + 4.37i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (9.81 - 3.18i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.321 - 0.989i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.69iT - 43T^{2} \)
47 \( 1 + (3.57 - 11.0i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.76 + 5.18i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.80 - 1.56i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.57 - 3.53i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.35iT - 67T^{2} \)
71 \( 1 + (-0.523 - 0.720i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (11.9 - 3.88i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.93 - 7.21i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.00 - 4.14i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.43iT - 89T^{2} \)
97 \( 1 + (-7.02 + 9.66i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.41018410377522301766295008465, −10.02453185780216556867855289123, −9.326192097670347394679949706880, −8.823635576010412556900772559864, −7.65303697838426572138460618052, −6.71485228493883341972345337869, −5.39627980789773862947895173561, −4.51388342843301151171787747257, −3.13269759759446054164486408668, −2.40030073021782909445710306131, 1.78902488264240272270342479965, 3.04209614391620350423199609255, 3.44609877281370844525800996169, 5.26515734107012826736693938335, 6.45862100182439846980749795347, 7.22337884066211206020584405941, 8.471799726047605034335336080208, 9.228715728388503638348181653097, 10.31698766766288891250072124752, 10.77082496860148661069734750742

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