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Properties

Label 2-308-11.9-c1-0-5
Degree $2$
Conductor $308$
Sign $-0.0665 + 0.997i$
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  + (1.06 − 0.774i)3-s + (−1.02 − 3.14i)5-s + (−0.809 − 0.587i)7-s + (−0.390 + 1.20i)9-s + (1.37 − 3.01i)11-s + (0.176 − 0.543i)13-s + (−3.52 − 2.55i)15-s + (−1.58 − 4.86i)17-s + (−4.67 + 3.39i)19-s − 1.31·21-s + 7.20·23-s + (−4.78 + 3.47i)25-s + (1.73 + 5.34i)27-s + (8.32 + 6.04i)29-s + (3.27 − 10.0i)31-s + ⋯
L(s)  = 1  + (0.615 − 0.447i)3-s + (−0.456 − 1.40i)5-s + (−0.305 − 0.222i)7-s + (−0.130 + 0.400i)9-s + (0.413 − 0.910i)11-s + (0.0489 − 0.150i)13-s + (−0.909 − 0.660i)15-s + (−0.383 − 1.18i)17-s + (−1.07 + 0.779i)19-s − 0.287·21-s + 1.50·23-s + (−0.957 + 0.695i)25-s + (0.334 + 1.02i)27-s + (1.54 + 1.12i)29-s + (0.588 − 1.80i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0665 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0665 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.0665 + 0.997i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ -0.0665 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.925787 - 0.989609i\)
\(L(\frac12)\) \(\approx\) \(0.925787 - 0.989609i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-1.37 + 3.01i)T \)
good3 \( 1 + (-1.06 + 0.774i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (1.02 + 3.14i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.176 + 0.543i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.58 + 4.86i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.67 - 3.39i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 7.20T + 23T^{2} \)
29 \( 1 + (-8.32 - 6.04i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.27 + 10.0i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.356 - 0.258i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.77 - 4.92i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.53T + 43T^{2} \)
47 \( 1 + (-8.06 + 5.86i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.98 - 9.20i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.66 - 1.20i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.87 - 8.85i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 1.05T + 67T^{2} \)
71 \( 1 + (1.77 + 5.45i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.27 - 4.55i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.123 - 0.381i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.72 - 5.29i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 2.15T + 89T^{2} \)
97 \( 1 + (1.13 - 3.50i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.61837485096018723237509813706, −10.57788935263738589208151623120, −9.161109484944450477318047400916, −8.617840623482793886080694295535, −7.88168493461257919962409352514, −6.69825642424203054034460991492, −5.30124714687268315369241928144, −4.27130319800943824845923999834, −2.82774979972432704286357424342, −0.990424287524757850984999146273, 2.51219885585835293846953793497, 3.48810825441523660810572961815, 4.53812245235677619306681822970, 6.54478562608303722404106137104, 6.79533880646196552622344742945, 8.298029801630079396418040277074, 9.097312018424379909270767796856, 10.19132909718504117448976209308, 10.80978757575445134238229109530, 11.88593151820929397078725714279

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