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Properties

Label 2-308-11.3-c1-0-2
Degree $2$
Conductor $308$
Sign $0.997 + 0.0744i$
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  + (0.153 + 0.471i)3-s + (−0.489 − 0.355i)5-s + (0.309 − 0.951i)7-s + (2.22 − 1.61i)9-s + (3.31 + 0.168i)11-s + (0.540 − 0.392i)13-s + (0.0928 − 0.285i)15-s + (4.47 + 3.25i)17-s + (0.917 + 2.82i)19-s + 0.495·21-s − 2.32·23-s + (−1.43 − 4.40i)25-s + (2.30 + 1.67i)27-s + (1.81 − 5.58i)29-s + (−5.67 + 4.12i)31-s + ⋯
L(s)  = 1  + (0.0884 + 0.272i)3-s + (−0.219 − 0.159i)5-s + (0.116 − 0.359i)7-s + (0.742 − 0.539i)9-s + (0.998 + 0.0507i)11-s + (0.149 − 0.108i)13-s + (0.0239 − 0.0737i)15-s + (1.08 + 0.789i)17-s + (0.210 + 0.647i)19-s + 0.108·21-s − 0.485·23-s + (−0.286 − 0.881i)25-s + (0.444 + 0.322i)27-s + (0.337 − 1.03i)29-s + (−1.01 + 0.740i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.997 + 0.0744i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.997 + 0.0744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43730 - 0.0536051i\)
\(L(\frac12)\) \(\approx\) \(1.43730 - 0.0536051i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 - 0.168i)T \)
good3 \( 1 + (-0.153 - 0.471i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.489 + 0.355i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-0.540 + 0.392i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.47 - 3.25i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.917 - 2.82i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.32T + 23T^{2} \)
29 \( 1 + (-1.81 + 5.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.67 - 4.12i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.54 - 7.82i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.71 + 11.4i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.96T + 43T^{2} \)
47 \( 1 + (0.388 + 1.19i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.12 + 2.99i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.59 - 11.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.14 + 3.74i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + (3.35 + 2.43i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.80 - 8.62i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.81 - 4.22i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.26 - 4.55i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.56T + 89T^{2} \)
97 \( 1 + (2.93 - 2.13i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.03930848833587876569151979387, −10.50804252977808268075258139421, −9.961427246368792262223000642348, −8.886193409816260238689499193186, −7.901013567660974274344023615994, −6.84030304788910483689422160802, −5.77507529086904580962004192551, −4.26495452440634767132339053738, −3.58638234501643186448300408490, −1.40857313914900957958651025578, 1.58360176413345681879884343814, 3.24942325025063202547615689966, 4.57131194574423046775001866353, 5.77862065628269372905253873797, 7.05945790105923532913030667007, 7.68963085644460660571152101325, 8.964068863587266725839658667964, 9.729057714540604748201948467280, 10.91059045336892143723836771466, 11.73677076950842829577332495147

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