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Properties

Label 2-414-23.8-c1-0-0
Degree $2$
Conductor $414$
Sign $-0.988 + 0.150i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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  + (−0.142 + 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.144 − 0.166i)5-s + (−2.90 − 1.86i)7-s + (0.415 − 0.909i)8-s + (0.185 − 0.119i)10-s + (0.752 + 5.23i)11-s + (−5.58 + 3.58i)13-s + (2.26 − 2.60i)14-s + (0.841 + 0.540i)16-s + (−4.80 + 1.41i)17-s + (−3.97 − 1.16i)19-s + (0.0915 + 0.200i)20-s − 5.29·22-s + (−1.48 + 4.56i)23-s + ⋯
L(s)  = 1  + (−0.100 + 0.699i)2-s + (−0.479 − 0.140i)4-s + (−0.0645 − 0.0745i)5-s + (−1.09 − 0.705i)7-s + (0.146 − 0.321i)8-s + (0.0586 − 0.0376i)10-s + (0.227 + 1.57i)11-s + (−1.54 + 0.994i)13-s + (0.604 − 0.697i)14-s + (0.210 + 0.135i)16-s + (−1.16 + 0.342i)17-s + (−0.911 − 0.267i)19-s + (0.0204 + 0.0448i)20-s − 1.12·22-s + (−0.308 + 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.988 + 0.150i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.988 + 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0262240 - 0.346056i\)
\(L(\frac12)\) \(\approx\) \(0.0262240 - 0.346056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 - 0.989i)T \)
3 \( 1 \)
23 \( 1 + (1.48 - 4.56i)T \)
good5 \( 1 + (0.144 + 0.166i)T + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (2.90 + 1.86i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (-0.752 - 5.23i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (5.58 - 3.58i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (4.80 - 1.41i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (3.97 + 1.16i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-9.00 + 2.64i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-0.780 + 1.70i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (0.918 - 1.06i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (5.30 + 6.12i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-3.51 - 7.70i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + 4.00T + 47T^{2} \)
53 \( 1 + (-5.79 - 3.72i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (10.8 - 6.95i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-5.14 + 11.2i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (0.339 - 2.36i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (-0.489 + 3.40i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (1.30 + 0.383i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (-0.440 + 0.283i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (4.44 - 5.13i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-4.40 - 9.65i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-6.92 - 7.99i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.93726110452663891925302016825, −10.40235915000790917565022312561, −9.791706082863042679912211787249, −9.098545471973605698314777136910, −7.79972915804213095082744624252, −6.78474640705915470493635606417, −6.55572635697213069575158823755, −4.70249542974119465499309510434, −4.20192794910188987148348461425, −2.28752226295585379074682616352, 0.21520548797017779712709461002, 2.56123024821089362607819700992, 3.26963925227228004639446708470, 4.77747420893257359310425196299, 5.93785520923853884426828660926, 6.89009965058905308392897536577, 8.386533033866364222414610241007, 8.916951978124734337699000151109, 10.00013424339798572471089586397, 10.65856168710058343243922741153

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