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Properties

Label 2-608-152.107-c1-0-12
Degree $2$
Conductor $608$
Sign $0.750 + 0.660i$
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  + (1.05 − 0.606i)3-s + (2.45 − 1.41i)5-s − 0.450i·7-s + (−0.763 + 1.32i)9-s + 2.15·11-s + (1.86 − 3.22i)13-s + (1.71 − 2.97i)15-s + (−0.716 − 1.24i)17-s + (0.0305 + 4.35i)19-s + (−0.273 − 0.473i)21-s + (−1.12 − 0.652i)23-s + (1.50 − 2.60i)25-s + 5.49i·27-s + (4.22 − 7.32i)29-s − 0.497·31-s + ⋯
L(s)  = 1  + (0.606 − 0.350i)3-s + (1.09 − 0.632i)5-s − 0.170i·7-s + (−0.254 + 0.440i)9-s + 0.651·11-s + (0.516 − 0.895i)13-s + (0.443 − 0.767i)15-s + (−0.173 − 0.301i)17-s + (0.00701 + 0.999i)19-s + (−0.0596 − 0.103i)21-s + (−0.235 − 0.136i)23-s + (0.300 − 0.520i)25-s + 1.05i·27-s + (0.785 − 1.36i)29-s − 0.0893·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.750 + 0.660i$
Analytic conductor: \(4.85490\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1/2),\ 0.750 + 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02085 - 0.763113i\)
\(L(\frac12)\) \(\approx\) \(2.02085 - 0.763113i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0305 - 4.35i)T \)
good3 \( 1 + (-1.05 + 0.606i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.45 + 1.41i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.450iT - 7T^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
13 \( 1 + (-1.86 + 3.22i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.716 + 1.24i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.12 + 0.652i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.22 + 7.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.497T + 31T^{2} \)
37 \( 1 + 6.72T + 37T^{2} \)
41 \( 1 + (7.30 - 4.21i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.90 + 5.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.567 + 0.327i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.86 - 6.69i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-12.1 + 6.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.13 - 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.16 + 1.25i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.35 - 14.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-8.25 - 14.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.05 + 7.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (6.95 + 4.01i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.47 - 3.15i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.27952819407245471978064712256, −9.722595434016033919888838480681, −8.582770500763588742697232633525, −8.233130206419761122290344572102, −6.98418291273406640703019149323, −5.90724097246027928312060559728, −5.18413101462716377110049729784, −3.77928916268675030297333586198, −2.44933019078936109189890896790, −1.35434683680982266005343196499, 1.77750824071507132069706157463, 2.94627510614988239868098123989, 3.94253816189995694350728757689, 5.28018609079044079919002659829, 6.48194536095137996244752181564, 6.81529919285566093340396134341, 8.507996719657522001948748894687, 9.030385864659516475490771913334, 9.756494878764156955002649805567, 10.59328856954602620312929930221

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