Properties

Label 2-418-19.9-c1-0-16
Degree $2$
Conductor $418$
Sign $0.172 + 0.984i$
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  + (0.173 + 0.984i)2-s + (2.40 − 2.01i)3-s + (−0.939 + 0.342i)4-s + (−3.99 − 1.45i)5-s + (2.40 + 2.01i)6-s + (0.394 − 0.683i)7-s + (−0.5 − 0.866i)8-s + (1.19 − 6.76i)9-s + (0.738 − 4.18i)10-s + (0.5 + 0.866i)11-s + (−1.57 + 2.72i)12-s + (−2.19 − 1.84i)13-s + (0.741 + 0.269i)14-s + (−12.5 + 4.56i)15-s + (0.766 − 0.642i)16-s + (−0.846 − 4.79i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (1.38 − 1.16i)3-s + (−0.469 + 0.171i)4-s + (−1.78 − 0.650i)5-s + (0.982 + 0.824i)6-s + (0.149 − 0.258i)7-s + (−0.176 − 0.306i)8-s + (0.397 − 2.25i)9-s + (0.233 − 1.32i)10-s + (0.150 + 0.261i)11-s + (−0.453 + 0.785i)12-s + (−0.608 − 0.510i)13-s + (0.198 + 0.0721i)14-s + (−3.24 + 1.17i)15-s + (0.191 − 0.160i)16-s + (−0.205 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.172 + 0.984i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.172 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15240 - 0.968017i\)
\(L(\frac12)\) \(\approx\) \(1.15240 - 0.968017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-4.07 - 1.55i)T \)
good3 \( 1 + (-2.40 + 2.01i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (3.99 + 1.45i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.394 + 0.683i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (2.19 + 1.84i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.846 + 4.79i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-2.82 + 1.02i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.17 + 6.69i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (5.14 - 8.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.927T + 37T^{2} \)
41 \( 1 + (0.0603 - 0.0506i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-4.02 - 1.46i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.192 + 1.09i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-1.06 + 0.389i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.434 - 2.46i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-6.64 + 2.41i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.00672 - 0.0381i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-13.9 - 5.08i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-6.96 + 5.84i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (1.07 - 0.903i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.53 - 9.58i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.408 + 0.342i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (2.07 + 11.7i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.37834370168870176791563824216, −9.572349361744670672717457036683, −8.731215547842616818712117000145, −8.031692720478566011829187319312, −7.37781616018020296039695163921, −6.99617145961790140071621157635, −5.06284637398654833663871534599, −3.92668838707536506031141958252, −2.94715931820720918219084542465, −0.842196448495279610990875725157, 2.48332328506297617532684705811, 3.53677369334722652535150365955, 3.98204049327876803633126707921, 5.02998948387969354764014330753, 7.16278829020018730024911545827, 8.028585616897371031839792457964, 8.771886129256676492189978623206, 9.555962951631781714126531164772, 10.66314549929894307457948219430, 11.16865570879089838696529301644

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