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Properties

Label 2-308-77.58-c1-0-1
Degree $2$
Conductor $308$
Sign $-0.276 - 0.960i$
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  + (0.599 − 0.665i)3-s + (−4.03 + 1.79i)5-s + (0.860 + 2.50i)7-s + (0.229 + 2.18i)9-s + (−0.394 − 3.29i)11-s + (−3.36 + 2.44i)13-s + (−1.22 + 3.76i)15-s + (−0.518 + 4.93i)17-s + (−0.672 − 0.142i)19-s + (2.18 + 0.927i)21-s + (−2.70 + 4.67i)23-s + (9.69 − 10.7i)25-s + (3.76 + 2.73i)27-s + (1.02 − 3.15i)29-s + (6.03 + 2.68i)31-s + ⋯
L(s)  = 1  + (0.346 − 0.384i)3-s + (−1.80 + 0.802i)5-s + (0.325 + 0.945i)7-s + (0.0765 + 0.728i)9-s + (−0.119 − 0.992i)11-s + (−0.933 + 0.678i)13-s + (−0.315 + 0.970i)15-s + (−0.125 + 1.19i)17-s + (−0.154 − 0.0327i)19-s + (0.475 + 0.202i)21-s + (−0.563 + 0.975i)23-s + (1.93 − 2.15i)25-s + (0.724 + 0.526i)27-s + (0.190 − 0.586i)29-s + (1.08 + 0.482i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.276 - 0.960i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ -0.276 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482827 + 0.641652i\)
\(L(\frac12)\) \(\approx\) \(0.482827 + 0.641652i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.860 - 2.50i)T \)
11 \( 1 + (0.394 + 3.29i)T \)
good3 \( 1 + (-0.599 + 0.665i)T + (-0.313 - 2.98i)T^{2} \)
5 \( 1 + (4.03 - 1.79i)T + (3.34 - 3.71i)T^{2} \)
13 \( 1 + (3.36 - 2.44i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.518 - 4.93i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (0.672 + 0.142i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (2.70 - 4.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.02 + 3.15i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.03 - 2.68i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (1.71 + 1.90i)T + (-3.86 + 36.7i)T^{2} \)
41 \( 1 + (1.30 + 4.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.76T + 43T^{2} \)
47 \( 1 + (3.55 + 0.754i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (1.29 + 0.577i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (5.60 - 1.19i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-5.37 + 2.39i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (-0.225 - 0.390i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.43 + 5.39i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.52 + 0.535i)T + (66.6 - 29.6i)T^{2} \)
79 \( 1 + (-1.14 - 10.8i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-5.68 - 4.13i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.82 + 3.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.83 + 3.51i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.85545303151644315525165039742, −11.25193644532896253934737983728, −10.36063944301749340500539060264, −8.730971394445857228624319266979, −8.083468820455602458220564159755, −7.43830233730852169366798547590, −6.26416741328062013466936521249, −4.75548752783105289121966232858, −3.53906823720121875625595734869, −2.34379219165138890184372935602, 0.56232238394200371749727788912, 3.16364014185718523159343262840, 4.44506133116627794108843203967, 4.69814580174613838770864530166, 6.94535814112030108345131760111, 7.65687017861867766200764831653, 8.433092894201859486241735598526, 9.544779404324327557017515294878, 10.43259742683773591774922521570, 11.64737010147357683969736685516

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