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Properties

Label 2-418-209.151-c1-0-12
Degree $2$
Conductor $418$
Sign $0.927 + 0.372i$
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 + (0.809 − 0.263i)3-s + (0.309 − 0.951i)4-s + (2.31 + 1.67i)5-s + (0.500 − 0.688i)6-s + (1.24 + 0.405i)7-s + (−0.309 − 0.951i)8-s + (−1.84 + 1.33i)9-s + 2.85·10-s + (2.44 − 2.24i)11-s − 0.851i·12-s + (−2.28 + 1.66i)13-s + (1.24 − 0.405i)14-s + (2.31 + 0.751i)15-s + (−0.809 − 0.587i)16-s + (−2.39 + 3.29i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.467 − 0.151i)3-s + (0.154 − 0.475i)4-s + (1.03 + 0.751i)5-s + (0.204 − 0.281i)6-s + (0.471 + 0.153i)7-s + (−0.109 − 0.336i)8-s + (−0.613 + 0.445i)9-s + 0.903·10-s + (0.737 − 0.675i)11-s − 0.245i·12-s + (−0.634 + 0.460i)13-s + (0.333 − 0.108i)14-s + (0.597 + 0.194i)15-s + (−0.202 − 0.146i)16-s + (−0.580 + 0.799i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.927 + 0.372i$
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: \((2,\ 418,\ (\ :1/2),\ 0.927 + 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43182 - 0.470368i\)
\(L(\frac12)\) \(\approx\) \(2.43182 - 0.470368i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 + 2.24i)T \)
19 \( 1 + (-2.33 + 3.68i)T \)
good3 \( 1 + (-0.809 + 0.263i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.31 - 1.67i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.24 - 0.405i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.28 - 1.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.39 - 3.29i)T + (-5.25 - 16.1i)T^{2} \)
23 \( 1 + 2.64T + 23T^{2} \)
29 \( 1 + (-1.30 + 4.00i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.00 + 5.51i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.80 - 0.586i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.72 + 11.4i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.0556iT - 43T^{2} \)
47 \( 1 + (-2.47 - 7.61i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.64 - 2.25i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.90 - 1.59i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.510 - 0.702i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.0iT - 67T^{2} \)
71 \( 1 + (7.58 - 10.4i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.29 + 2.69i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.96 - 4.33i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.21 + 7.17i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.88iT - 89T^{2} \)
97 \( 1 + (-7.58 - 10.4i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.24832791233260777458920365012, −10.38934438905285897454878952452, −9.402098680848866145722797189287, −8.585598996060964409374383402105, −7.29198340917300314078192668905, −6.21760465738176370371082557359, −5.47757271709267535032907939567, −4.08908572319176275615785210687, −2.70595806131744910499021874951, −1.96213799063112154338475802776, 1.78804784935741848896239552865, 3.23096882980618386841823152983, 4.61537870768221620042340621531, 5.38333456558284284456744935085, 6.40789923926631306368199297893, 7.52626523052716343715491431078, 8.597840981174643748272652116809, 9.341147553997982076254430671972, 10.09379281501751228639599619439, 11.54721462034638610525675423263

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