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Properties

Label 2-418-209.18-c1-0-14
Degree $2$
Conductor $418$
Sign $0.981 + 0.191i$
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 + (−0.819 − 0.266i)3-s + (0.309 + 0.951i)4-s + (2.09 − 1.52i)5-s + (−0.506 − 0.696i)6-s + (3.24 − 1.05i)7-s + (−0.309 + 0.951i)8-s + (−1.82 − 1.32i)9-s + 2.59·10-s + (−2.10 − 2.56i)11-s − 0.861i·12-s + (0.159 + 0.115i)13-s + (3.24 + 1.05i)14-s + (−2.12 + 0.689i)15-s + (−0.809 + 0.587i)16-s + (2.24 + 3.08i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.472 − 0.153i)3-s + (0.154 + 0.475i)4-s + (0.937 − 0.680i)5-s + (−0.206 − 0.284i)6-s + (1.22 − 0.398i)7-s + (−0.109 + 0.336i)8-s + (−0.608 − 0.442i)9-s + 0.819·10-s + (−0.634 − 0.773i)11-s − 0.248i·12-s + (0.0441 + 0.0321i)13-s + (0.867 + 0.281i)14-s + (−0.547 + 0.178i)15-s + (−0.202 + 0.146i)16-s + (0.543 + 0.748i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.981 + 0.191i$
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: \((2,\ 418,\ (\ :1/2),\ 0.981 + 0.191i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94669 - 0.188345i\)
\(L(\frac12)\) \(\approx\) \(1.94669 - 0.188345i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 + 2.56i)T \)
19 \( 1 + (0.573 + 4.32i)T \)
good3 \( 1 + (0.819 + 0.266i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.09 + 1.52i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-3.24 + 1.05i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.159 - 0.115i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.24 - 3.08i)T + (-5.25 + 16.1i)T^{2} \)
23 \( 1 - 4.85T + 23T^{2} \)
29 \( 1 + (-2.87 - 8.84i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.804 + 1.10i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (7.66 - 2.49i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.38 + 10.4i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.88iT - 43T^{2} \)
47 \( 1 + (-0.153 + 0.471i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.23 - 9.95i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.02 - 0.657i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.13 + 1.55i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 4.77iT - 67T^{2} \)
71 \( 1 + (4.08 + 5.61i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.64 - 3.13i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (12.6 + 9.19i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.36 - 7.38i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 14.4iT - 89T^{2} \)
97 \( 1 + (-4.34 + 5.98i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.11576479546712575729619528946, −10.65744696728653536727327098791, −9.043956165786825086993315901234, −8.483715586526961250967962042166, −7.33804039913975495318931063800, −6.17327297767583889482095270144, −5.35402206979229995285087077635, −4.77757206426753506884925901142, −3.09711796291867423036570023366, −1.32629364812305539178135460879, 1.90621834932224240575241403656, 2.83086675308060141084929235971, 4.66351216104169298574488965547, 5.35991204302307725567492707828, 6.13003371067558912633060779690, 7.43828654319469512625019306698, 8.487202321805888082316120846937, 9.914789442511576219430035727716, 10.38431794082862616466845608922, 11.34169350154761966404744738228

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