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Properties

Label 2-608-152.27-c1-0-4
Degree $2$
Conductor $608$
Sign $0.0172 - 0.999i$
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  + (2.65 + 1.53i)3-s + (−2.25 − 1.30i)5-s + 4.30i·7-s + (3.20 + 5.54i)9-s + 0.349·11-s + (0.839 + 1.45i)13-s + (−3.99 − 6.92i)15-s + (0.357 − 0.618i)17-s + (−4.35 − 0.268i)19-s + (−6.59 + 11.4i)21-s + (1.38 − 0.797i)23-s + (0.895 + 1.55i)25-s + 10.4i·27-s + (0.463 + 0.803i)29-s + 2.80·31-s + ⋯
L(s)  = 1  + (1.53 + 0.885i)3-s + (−1.00 − 0.582i)5-s + 1.62i·7-s + (1.06 + 1.84i)9-s + 0.105·11-s + (0.232 + 0.403i)13-s + (−1.03 − 1.78i)15-s + (0.0865 − 0.149i)17-s + (−0.998 − 0.0615i)19-s + (−1.43 + 2.49i)21-s + (0.287 − 0.166i)23-s + (0.179 + 0.310i)25-s + 2.00i·27-s + (0.0860 + 0.149i)29-s + 0.503·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0172 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0172 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.0172 - 0.999i$
Analytic conductor: \(4.85490\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1/2),\ 0.0172 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43820 + 1.41361i\)
\(L(\frac12)\) \(\approx\) \(1.43820 + 1.41361i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.35 + 0.268i)T \)
good3 \( 1 + (-2.65 - 1.53i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.25 + 1.30i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.30iT - 7T^{2} \)
11 \( 1 - 0.349T + 11T^{2} \)
13 \( 1 + (-0.839 - 1.45i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.357 + 0.618i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.38 + 0.797i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.463 - 0.803i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.80T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + (-4.87 - 2.81i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.32 - 7.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.26 + 3.61i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.73 + 8.20i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.62 + 3.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.46 - 2.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.6 + 6.12i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.09 + 8.82i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.54 + 4.40i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.31 + 2.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.69T + 83T^{2} \)
89 \( 1 + (-7.37 + 4.25i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.30 - 3.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.81157361637825377342531455106, −9.474571910037670618675045546084, −9.165672975836866202991192257263, −8.215327239260751407409934381610, −8.020724833079469235288407885113, −6.35595245124489109467996560015, −4.91595510772620923939479369532, −4.23161297071686565261170022498, −3.14304972278818695917504914591, −2.18502660608528503568115913037, 1.00785052251077847161472143239, 2.62380273248191343793938389039, 3.68914862708436636649918105405, 4.19474003809256161304421644936, 6.41194022892123764924386534274, 7.22535899534249062377815008177, 7.73000693804715028802136314914, 8.330231442827516062282250934603, 9.420468127689507135988716256128, 10.47565413003531863922469986961

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