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Properties

Label 2-608-19.9-c1-0-2
Degree $2$
Conductor $608$
Sign $0.187 - 0.982i$
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  + (−0.990 + 0.830i)3-s + (−3.16 − 1.15i)5-s + (1.52 − 2.64i)7-s + (−0.230 + 1.30i)9-s + (−0.584 − 1.01i)11-s + (2.63 + 2.20i)13-s + (4.08 − 1.48i)15-s + (0.907 + 5.14i)17-s + (−2.29 + 3.70i)19-s + (0.686 + 3.89i)21-s + (5.21 − 1.89i)23-s + (4.85 + 4.07i)25-s + (−2.79 − 4.84i)27-s + (−1.32 + 7.52i)29-s + (−2.02 + 3.51i)31-s + ⋯
L(s)  = 1  + (−0.571 + 0.479i)3-s + (−1.41 − 0.514i)5-s + (0.577 − 1.00i)7-s + (−0.0769 + 0.436i)9-s + (−0.176 − 0.305i)11-s + (0.730 + 0.612i)13-s + (1.05 − 0.384i)15-s + (0.220 + 1.24i)17-s + (−0.525 + 0.850i)19-s + (0.149 + 0.849i)21-s + (1.08 − 0.395i)23-s + (0.970 + 0.814i)25-s + (−0.538 − 0.932i)27-s + (−0.246 + 1.39i)29-s + (−0.364 + 0.631i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(4.85490\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1/2),\ 0.187 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.601194 + 0.497427i\)
\(L(\frac12)\) \(\approx\) \(0.601194 + 0.497427i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.29 - 3.70i)T \)
good3 \( 1 + (0.990 - 0.830i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (3.16 + 1.15i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.52 + 2.64i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.584 + 1.01i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.63 - 2.20i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.907 - 5.14i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-5.21 + 1.89i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.32 - 7.52i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.02 - 3.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.08T + 37T^{2} \)
41 \( 1 + (4.06 - 3.41i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-9.97 - 3.62i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.643 - 3.64i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (7.94 - 2.89i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.18 + 6.70i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-10.1 + 3.69i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.96 - 11.1i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (0.135 + 0.0494i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (4.23 - 3.55i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-11.3 + 9.50i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-4.31 + 7.47i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.10 + 0.929i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.97 - 11.1i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.97004763860625750042223380408, −10.38233212522700458610307774673, −8.879372022692038782445239655380, −8.144556577174588539224693138809, −7.51831700492980639928504776481, −6.29764966325261349430472507311, −5.03307370325781329467660729521, −4.28837188169866157987026354583, −3.59542888745730582237876527496, −1.28144144646112098516240365410, 0.53805307616033626104370019192, 2.57440691414156349299954266604, 3.72483141587694356814242635313, 4.98114391752804729613945810011, 5.92666649051299035274777697394, 7.01816621328970350754457540813, 7.65342822312583758583575927519, 8.589334983874769181616936364015, 9.455404911643525719053853116139, 10.97095050814395725086475330170

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