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Properties

Label 2-308-11.3-c1-0-2
Degree 22
Conductor 308308
Sign 0.997+0.0744i0.997 + 0.0744i
Analytic cond. 2.459392.45939
Root an. cond. 1.568241.56824
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(308s/2 ΓC(s) L(s)=((0.997+0.0744i) Λ‾(2−s)\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}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((0.997+0.0744i) Λ‾(1−s)\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: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: 0.997+0.0744i0.997 + 0.0744i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(113,⋅)\chi_{308} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), 0.997+0.0744i)(2,\ 308,\ (\ :1/2),\ 0.997 + 0.0744i)

Particular Values

L(1)L(1) ≈\approx 1.43730−0.0536051i1.43730 - 0.0536051i
L(12)L(\frac12) ≈\approx 1.43730−0.0536051i1.43730 - 0.0536051i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1+(−0.309+0.951i)T 1 + (-0.309 + 0.951i)T
11 1+(−3.31−0.168i)T 1 + (-3.31 - 0.168i)T
good3 1+(−0.153−0.471i)T+(−2.42+1.76i)T2 1 + (-0.153 - 0.471i)T + (-2.42 + 1.76i)T^{2}
5 1+(0.489+0.355i)T+(1.54+4.75i)T2 1 + (0.489 + 0.355i)T + (1.54 + 4.75i)T^{2}
13 1+(−0.540+0.392i)T+(4.01−12.3i)T2 1 + (-0.540 + 0.392i)T + (4.01 - 12.3i)T^{2}
17 1+(−4.47−3.25i)T+(5.25+16.1i)T2 1 + (-4.47 - 3.25i)T + (5.25 + 16.1i)T^{2}
19 1+(−0.917−2.82i)T+(−15.3+11.1i)T2 1 + (-0.917 - 2.82i)T + (-15.3 + 11.1i)T^{2}
23 1+2.32T+23T2 1 + 2.32T + 23T^{2}
29 1+(−1.81+5.58i)T+(−23.4−17.0i)T2 1 + (-1.81 + 5.58i)T + (-23.4 - 17.0i)T^{2}
31 1+(5.67−4.12i)T+(9.57−29.4i)T2 1 + (5.67 - 4.12i)T + (9.57 - 29.4i)T^{2}
37 1+(2.54−7.82i)T+(−29.9−21.7i)T2 1 + (2.54 - 7.82i)T + (-29.9 - 21.7i)T^{2}
41 1+(3.71+11.4i)T+(−33.1+24.0i)T2 1 + (3.71 + 11.4i)T + (-33.1 + 24.0i)T^{2}
43 1−2.96T+43T2 1 - 2.96T + 43T^{2}
47 1+(0.388+1.19i)T+(−38.0+27.6i)T2 1 + (0.388 + 1.19i)T + (-38.0 + 27.6i)T^{2}
53 1+(−4.12+2.99i)T+(16.3−50.4i)T2 1 + (-4.12 + 2.99i)T + (16.3 - 50.4i)T^{2}
59 1+(3.59−11.0i)T+(−47.7−34.6i)T2 1 + (3.59 - 11.0i)T + (-47.7 - 34.6i)T^{2}
61 1+(5.14+3.74i)T+(18.8+58.0i)T2 1 + (5.14 + 3.74i)T + (18.8 + 58.0i)T^{2}
67 1+13.4T+67T2 1 + 13.4T + 67T^{2}
71 1+(3.35+2.43i)T+(21.9+67.5i)T2 1 + (3.35 + 2.43i)T + (21.9 + 67.5i)T^{2}
73 1+(2.80−8.62i)T+(−59.0−42.9i)T2 1 + (2.80 - 8.62i)T + (-59.0 - 42.9i)T^{2}
79 1+(5.81−4.22i)T+(24.4−75.1i)T2 1 + (5.81 - 4.22i)T + (24.4 - 75.1i)T^{2}
83 1+(−6.26−4.55i)T+(25.6+78.9i)T2 1 + (-6.26 - 4.55i)T + (25.6 + 78.9i)T^{2}
89 1+2.56T+89T2 1 + 2.56T + 89T^{2}
97 1+(2.93−2.13i)T+(29.9−92.2i)T2 1 + (2.93 - 2.13i)T + (29.9 - 92.2i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line