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Properties

Label 2-608-152.149-c1-0-10
Degree $2$
Conductor $608$
Sign $0.999 + 0.0383i$
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  + (0.467 + 1.28i)3-s + (1.73 + 0.305i)5-s + (−1.91 − 3.31i)7-s + (0.868 − 0.728i)9-s + (2.26 + 1.30i)11-s + (2.03 − 5.60i)13-s + (0.417 + 2.36i)15-s + (2.13 + 1.78i)17-s + (−0.713 − 4.30i)19-s + (3.36 − 4.00i)21-s + (1.10 + 6.27i)23-s + (−1.78 − 0.650i)25-s + (4.89 + 2.82i)27-s + (2.78 + 3.31i)29-s + (−2.41 − 4.18i)31-s + ⋯
L(s)  = 1  + (0.269 + 0.741i)3-s + (0.775 + 0.136i)5-s + (−0.723 − 1.25i)7-s + (0.289 − 0.242i)9-s + (0.682 + 0.394i)11-s + (0.565 − 1.55i)13-s + (0.107 + 0.611i)15-s + (0.516 + 0.433i)17-s + (−0.163 − 0.986i)19-s + (0.733 − 0.873i)21-s + (0.230 + 1.30i)23-s + (−0.357 − 0.130i)25-s + (0.941 + 0.543i)27-s + (0.516 + 0.615i)29-s + (−0.433 − 0.751i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.999 + 0.0383i$
Analytic conductor: \(4.85490\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1/2),\ 0.999 + 0.0383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85355 - 0.0355500i\)
\(L(\frac12)\) \(\approx\) \(1.85355 - 0.0355500i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.713 + 4.30i)T \)
good3 \( 1 + (-0.467 - 1.28i)T + (-2.29 + 1.92i)T^{2} \)
5 \( 1 + (-1.73 - 0.305i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (1.91 + 3.31i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.26 - 1.30i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.03 + 5.60i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.13 - 1.78i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.10 - 6.27i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-2.78 - 3.31i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.41 + 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.50iT - 37T^{2} \)
41 \( 1 + (6.78 - 2.46i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-3.24 - 0.571i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (3.73 - 3.13i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.49 - 0.264i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-4.81 + 5.74i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (5.64 - 0.995i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.11 - 2.51i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.236 - 1.34i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-10.5 + 3.83i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-6.51 + 2.37i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-5.89 + 3.40i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.91 + 2.51i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (9.33 + 7.83i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.32128389995216536242928382796, −9.883957287559254225031825606771, −9.259251902671235683913494775648, −8.014518412098663954789449415786, −6.96933056456489475924835527325, −6.19479858825696052477681870604, −5.00066192659907532106097801372, −3.80781669425754398586915207427, −3.17975842180787550727962625143, −1.19244251688227842013860007221, 1.60621375120401434171730072160, 2.47198735577228548716324821541, 3.91634703537960890848474097143, 5.37346469554205283789108979933, 6.33211178600282157680475997018, 6.78283968233124650007150998834, 8.170942062612025659561664743914, 8.982256780625914401696412652847, 9.524087217122482556026267449574, 10.56097741721647597329101227121

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