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Properties

Label 2-308-11.4-c1-0-1
Degree $2$
Conductor $308$
Sign $0.144 - 0.989i$
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  + (0.118 − 0.363i)3-s + (−2.78 + 2.02i)5-s + (−0.309 − 0.951i)7-s + (2.30 + 1.67i)9-s + (−0.555 + 3.26i)11-s + (4.59 + 3.33i)13-s + (0.405 + 1.24i)15-s + (−6.11 + 4.44i)17-s + (−0.347 + 1.06i)19-s − 0.381·21-s + 2.38·23-s + (2.10 − 6.48i)25-s + (1.80 − 1.31i)27-s + (−1.08 − 3.33i)29-s + (−4.30 − 3.13i)31-s + ⋯
L(s)  = 1  + (0.0681 − 0.209i)3-s + (−1.24 + 0.903i)5-s + (−0.116 − 0.359i)7-s + (0.769 + 0.559i)9-s + (−0.167 + 0.985i)11-s + (1.27 + 0.925i)13-s + (0.104 + 0.322i)15-s + (−1.48 + 1.07i)17-s + (−0.0797 + 0.245i)19-s − 0.0833·21-s + 0.496·23-s + (0.421 − 1.29i)25-s + (0.348 − 0.252i)27-s + (−0.201 − 0.619i)29-s + (−0.773 − 0.562i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.144 - 0.989i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749768 + 0.648531i\)
\(L(\frac12)\) \(\approx\) \(0.749768 + 0.648531i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.555 - 3.26i)T \)
good3 \( 1 + (-0.118 + 0.363i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.78 - 2.02i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-4.59 - 3.33i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.11 - 4.44i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.347 - 1.06i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.38T + 23T^{2} \)
29 \( 1 + (1.08 + 3.33i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.30 + 3.13i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.903 - 2.77i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.118 - 0.363i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.36T + 43T^{2} \)
47 \( 1 + (-2.99 + 9.21i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.57 + 4.04i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.08 - 12.5i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.41 - 5.38i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 + (-5.23 + 3.80i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.16 + 6.65i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.6 + 7.73i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.07 + 3.68i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 9.39T + 89T^{2} \)
97 \( 1 + (-3.98 - 2.89i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.74033019254042321591747468287, −10.95245548942273297816137590358, −10.36736267993771539544480753618, −8.972883712594890525056598601829, −7.87648055024285265613942264017, −7.12657946631831551720455615721, −6.39645286903700353259097596830, −4.38554550107433165431354905331, −3.83842114831796032285891319379, −2.02602249489111176250642275591, 0.75568738314087365910076947775, 3.23401548849196750650886278894, 4.20951206217435102244612120216, 5.31081076464309812978989564883, 6.65047730574904036251585589379, 7.82338663434439224754559812556, 8.767498054581380289149735504900, 9.239057002385318875838783132891, 10.94658336742741169258430781405, 11.26922401538192128219428532997

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