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Properties

Label 2-414-23.4-c1-0-0
Degree $2$
Conductor $414$
Sign $0.0508 - 0.998i$
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  + (−0.415 − 0.909i)2-s + (−0.654 + 0.755i)4-s + (−3.57 − 2.29i)5-s + (0.0421 + 0.293i)7-s + (0.959 + 0.281i)8-s + (−0.604 + 4.20i)10-s + (−0.995 + 2.18i)11-s + (−0.357 + 2.48i)13-s + (0.249 − 0.160i)14-s + (−0.142 − 0.989i)16-s + (2.95 + 3.40i)17-s + (−0.743 + 0.858i)19-s + (4.07 − 1.19i)20-s + 2.39·22-s + (−4.41 + 1.87i)23-s + ⋯
L(s)  = 1  + (−0.293 − 0.643i)2-s + (−0.327 + 0.377i)4-s + (−1.59 − 1.02i)5-s + (0.0159 + 0.110i)7-s + (0.339 + 0.0996i)8-s + (−0.191 + 1.32i)10-s + (−0.300 + 0.657i)11-s + (−0.0992 + 0.690i)13-s + (0.0666 − 0.0428i)14-s + (−0.0355 − 0.247i)16-s + (0.716 + 0.826i)17-s + (−0.170 + 0.196i)19-s + (0.911 − 0.267i)20-s + 0.511·22-s + (−0.920 + 0.391i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0508 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0508 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.0508 - 0.998i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.0508 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.199639 + 0.189730i\)
\(L(\frac12)\) \(\approx\) \(0.199639 + 0.189730i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 \)
23 \( 1 + (4.41 - 1.87i)T \)
good5 \( 1 + (3.57 + 2.29i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-0.0421 - 0.293i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (0.995 - 2.18i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.357 - 2.48i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-2.95 - 3.40i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (0.743 - 0.858i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-0.626 - 0.723i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (7.91 + 2.32i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (7.02 - 4.51i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (3.45 + 2.22i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-3.70 + 1.08i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 3.35T + 47T^{2} \)
53 \( 1 + (0.689 + 4.79i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (1.60 - 11.1i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (7.45 + 2.18i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (5.85 + 12.8i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-6.13 - 13.4i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-7.32 + 8.45i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (1.71 - 11.9i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (1.78 - 1.14i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (11.0 - 3.23i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (0.669 + 0.430i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.59295085857413500819132061425, −10.65458070297181200515534559640, −9.582618386875394653415770296785, −8.656002734999241299906032663513, −7.971888697310413307406084159901, −7.16579130555568160831598101063, −5.37502458831436554908375006965, −4.29031973787987613091043272356, −3.58683626033204419153547358580, −1.66304648671870163938997282622, 0.19871948082466420919876044394, 2.99018965869655882190846005041, 3.95576531728734970030107314981, 5.30374695332550117463832329446, 6.52613651740456378843292284613, 7.50344104714637820802062031973, 7.898319210735407605080357154854, 8.925888612953503318467268799153, 10.30933758255407748286237565858, 10.85289212617630230279054340410

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