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Properties

Label 2-308-77.37-c1-0-3
Degree $2$
Conductor $308$
Sign $0.993 - 0.114i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
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  + (−3.17 + 0.675i)3-s + (−0.0774 − 0.736i)5-s + (−2.58 − 0.558i)7-s + (6.91 − 3.07i)9-s + (2.92 + 1.56i)11-s + (−0.656 − 0.476i)13-s + (0.743 + 2.28i)15-s + (3.08 + 1.37i)17-s + (5.01 + 5.56i)19-s + (8.59 + 0.0278i)21-s + (2.39 − 4.15i)23-s + (4.35 − 0.925i)25-s + (−12.0 + 8.72i)27-s + (−0.829 − 2.55i)29-s + (0.647 − 6.15i)31-s + ⋯
L(s)  = 1  + (−1.83 + 0.390i)3-s + (−0.0346 − 0.329i)5-s + (−0.977 − 0.211i)7-s + (2.30 − 1.02i)9-s + (0.881 + 0.472i)11-s + (−0.182 − 0.132i)13-s + (0.192 + 0.591i)15-s + (0.749 + 0.333i)17-s + (1.14 + 1.27i)19-s + (1.87 + 0.00607i)21-s + (0.500 − 0.866i)23-s + (0.870 − 0.185i)25-s + (−2.31 + 1.67i)27-s + (−0.154 − 0.474i)29-s + (0.116 − 1.10i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.993 - 0.114i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.683419 + 0.0393356i\)
\(L(\frac12)\) \(\approx\) \(0.683419 + 0.0393356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (2.58 + 0.558i)T \)
11 \( 1 + (-2.92 - 1.56i)T \)
good3 \( 1 + (3.17 - 0.675i)T + (2.74 - 1.22i)T^{2} \)
5 \( 1 + (0.0774 + 0.736i)T + (-4.89 + 1.03i)T^{2} \)
13 \( 1 + (0.656 + 0.476i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.08 - 1.37i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (-5.01 - 5.56i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (-2.39 + 4.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.829 + 2.55i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.647 + 6.15i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-0.728 - 0.154i)T + (33.8 + 15.0i)T^{2} \)
41 \( 1 + (1.27 - 3.92i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.67T + 43T^{2} \)
47 \( 1 + (4.57 + 5.07i)T + (-4.91 + 46.7i)T^{2} \)
53 \( 1 + (1.39 - 13.2i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (-2.45 + 2.72i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (0.815 + 7.76i)T + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (-0.461 - 0.799i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.16 - 4.47i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-7.74 + 8.59i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (-5.42 + 2.41i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-2.53 + 1.84i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.944 - 1.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.58 - 1.88i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.90023447514093933123328545937, −10.75782219915475688088309704790, −9.969605626701363599128430735143, −9.377128595841449648493770386932, −7.57366795056636421477432180171, −6.50047895759757133342519664251, −5.85624858590160996358573070367, −4.74391009192541057866147657444, −3.70474613363436881415324607354, −0.950217109299280396261068451585, 0.969440719216643694717564809344, 3.29327029775518824755646611223, 4.96312378677144842582999587386, 5.77757448679329340272448370044, 6.82530272009966497539519379959, 7.19893464821038661019235610205, 9.128438407330201755203530700645, 9.994666839897055098054264047839, 11.04514631525474607239237080943, 11.61695125301035554387110242378

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