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Properties

Label 2-608-152.27-c1-0-15
Degree $2$
Conductor $608$
Sign $-0.669 + 0.742i$
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  + (−0.179 − 0.103i)3-s + (0.520 + 0.300i)5-s − 4.27i·7-s + (−1.47 − 2.56i)9-s − 5.11·11-s + (1.55 + 2.68i)13-s + (−0.0621 − 0.107i)15-s + (−1.52 + 2.64i)17-s + (−3.31 − 2.83i)19-s + (−0.442 + 0.765i)21-s + (−4.65 + 2.68i)23-s + (−2.31 − 4.01i)25-s + 1.23i·27-s + (−3.77 − 6.53i)29-s + 9.59·31-s + ⋯
L(s)  = 1  + (−0.103 − 0.0597i)3-s + (0.232 + 0.134i)5-s − 1.61i·7-s + (−0.492 − 0.853i)9-s − 1.54·11-s + (0.430 + 0.745i)13-s + (−0.0160 − 0.0277i)15-s + (−0.371 + 0.642i)17-s + (−0.759 − 0.650i)19-s + (−0.0964 + 0.167i)21-s + (−0.969 + 0.559i)23-s + (−0.463 − 0.803i)25-s + 0.237i·27-s + (−0.701 − 1.21i)29-s + 1.72·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $-0.669 + 0.742i$
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: \((2,\ 608,\ (\ :1/2),\ -0.669 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332456 - 0.747852i\)
\(L(\frac12)\) \(\approx\) \(0.332456 - 0.747852i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 + 2.83i)T \)
good3 \( 1 + (0.179 + 0.103i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.520 - 0.300i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.27iT - 7T^{2} \)
11 \( 1 + 5.11T + 11T^{2} \)
13 \( 1 + (-1.55 - 2.68i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.52 - 2.64i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.65 - 2.68i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.77 + 6.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.59T + 31T^{2} \)
37 \( 1 - 4.44T + 37T^{2} \)
41 \( 1 + (0.253 + 0.146i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.892 + 1.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.79 + 3.34i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.16 + 8.94i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.849 - 0.490i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.08 + 2.93i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.98 - 3.45i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.173 - 0.300i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.61 + 7.99i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.63 - 6.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.38T + 83T^{2} \)
89 \( 1 + (-5.44 + 3.14i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.87 - 4.54i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.32733145143064856988603070652, −9.701385941162533733283390362255, −8.415829477135248978550830165335, −7.71607867620793055756459275184, −6.62755302083272087518925661121, −5.98917330237096579064490394771, −4.52392670207257086522480565329, −3.75101999818955606995648729791, −2.28117533425056017096788160917, −0.42543600275817084739360916375, 2.20680548360523759286929587434, 2.90110830910986330030680666502, 4.75436937693080175788703455086, 5.57304332863625095243209792652, 6.06852737520597504379988776735, 7.75592462273385497996171233399, 8.292257897764895174078901807574, 9.116155595259266514535928490125, 10.22280873130424300106690053724, 10.87709775970042914384552811946

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