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Properties

Label 2-608-152.101-c1-0-12
Degree $2$
Conductor $608$
Sign $0.923 + 0.384i$
Analytic cond. $4.85490$
Root an. cond. $2.20338$
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  + (−0.945 + 2.59i)3-s + (−1.15 + 0.202i)5-s + (2.48 − 4.29i)7-s + (−3.56 − 2.98i)9-s + (1.12 − 0.651i)11-s + (−1.56 − 4.30i)13-s + (0.561 − 3.18i)15-s + (2.55 − 2.14i)17-s + (0.369 − 4.34i)19-s + (8.82 + 10.5i)21-s + (0.0598 − 0.339i)23-s + (−3.41 + 1.24i)25-s + (3.95 − 2.28i)27-s + (0.903 − 1.07i)29-s + (−2.59 + 4.49i)31-s + ⋯
L(s)  = 1  + (−0.546 + 1.50i)3-s + (−0.514 + 0.0907i)5-s + (0.937 − 1.62i)7-s + (−1.18 − 0.996i)9-s + (0.340 − 0.196i)11-s + (−0.434 − 1.19i)13-s + (0.144 − 0.821i)15-s + (0.619 − 0.520i)17-s + (0.0848 − 0.996i)19-s + (1.92 + 2.29i)21-s + (0.0124 − 0.0707i)23-s + (−0.683 + 0.248i)25-s + (0.760 − 0.438i)27-s + (0.167 − 0.199i)29-s + (−0.465 + 0.806i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.923 + 0.384i$
Analytic conductor: \(4.85490\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1/2),\ 0.923 + 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03637 - 0.207148i\)
\(L(\frac12)\) \(\approx\) \(1.03637 - 0.207148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-0.369 + 4.34i)T \)
good3 \( 1 + (0.945 - 2.59i)T + (-2.29 - 1.92i)T^{2} \)
5 \( 1 + (1.15 - 0.202i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.48 + 4.29i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.12 + 0.651i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.56 + 4.30i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.55 + 2.14i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.0598 + 0.339i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.903 + 1.07i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.59 - 4.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.14iT - 37T^{2} \)
41 \( 1 + (-8.84 - 3.22i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-6.63 + 1.17i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.689 - 0.578i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.26 + 0.223i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-6.06 - 7.23i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.91 + 0.513i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.95 + 2.33i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.96 + 11.1i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (1.39 + 0.507i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (8.75 + 3.18i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (11.1 + 6.46i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.44 + 0.889i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (2.82 - 2.36i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.64800884058676262128057417016, −10.00405246462267504015618606642, −9.064772933150066051433281900343, −7.81081894293809937209418291371, −7.25404262077392625081202627468, −5.70117239697536277906408813635, −4.79479440733866484493532660574, −4.15687979456163322148661387475, −3.23558456778545528524364449337, −0.67326675723615377510700563531, 1.54005542712239092090471365835, 2.33774306563399553735084656388, 4.23378769487846353466149849118, 5.56384036778219847280396815758, 6.07981687189075964508291608149, 7.23955768166890096232828394765, 7.956690679606242030229121935110, 8.655071502256976819014359216480, 9.704346808636173606668755653102, 11.36616105219564668128998402963

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