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Properties

Label 2-315-105.2-c1-0-1
Degree 22
Conductor 315315
Sign 0.789−0.613i0.789 - 0.613i
Analytic cond. 2.515282.51528
Root an. cond. 1.585961.58596
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  + (−0.543 − 2.02i)2-s + (−2.08 + 1.20i)4-s + (−1.61 + 1.54i)5-s + (1.07 + 2.41i)7-s + (0.609 + 0.609i)8-s + (4.01 + 2.43i)10-s + (−4.56 + 2.63i)11-s + (−3.08 + 3.08i)13-s + (4.31 − 3.50i)14-s + (−1.50 + 2.60i)16-s + (5.92 + 1.58i)17-s + (0.715 + 0.413i)19-s + (1.51 − 5.17i)20-s + (7.82 + 7.82i)22-s + (1.33 − 0.359i)23-s + ⋯
L(s)  = 1  + (−0.384 − 1.43i)2-s + (−1.04 + 0.602i)4-s + (−0.722 + 0.691i)5-s + (0.407 + 0.913i)7-s + (0.215 + 0.215i)8-s + (1.26 + 0.771i)10-s + (−1.37 + 0.794i)11-s + (−0.855 + 0.855i)13-s + (1.15 − 0.936i)14-s + (−0.376 + 0.651i)16-s + (1.43 + 0.385i)17-s + (0.164 + 0.0948i)19-s + (0.337 − 1.15i)20-s + (1.66 + 1.66i)22-s + (0.279 − 0.0748i)23-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=((0.789−0.613i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+1/2) L(s)=((0.789−0.613i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: 0.789−0.613i0.789 - 0.613i
Analytic conductor: 2.515282.51528
Root analytic conductor: 1.585961.58596
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ315(107,⋅)\chi_{315} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 315, ( :1/2), 0.789−0.613i)(2,\ 315,\ (\ :1/2),\ 0.789 - 0.613i)

Particular Values

L(1)L(1) ≈\approx 0.528315+0.181275i0.528315 + 0.181275i
L(12)L(\frac12) ≈\approx 0.528315+0.181275i0.528315 + 0.181275i
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)
bad3 1 1
5 1+(1.61−1.54i)T 1 + (1.61 - 1.54i)T
7 1+(−1.07−2.41i)T 1 + (-1.07 - 2.41i)T
good2 1+(0.543+2.02i)T+(−1.73+i)T2 1 + (0.543 + 2.02i)T + (-1.73 + i)T^{2}
11 1+(4.56−2.63i)T+(5.5−9.52i)T2 1 + (4.56 - 2.63i)T + (5.5 - 9.52i)T^{2}
13 1+(3.08−3.08i)T−13iT2 1 + (3.08 - 3.08i)T - 13iT^{2}
17 1+(−5.92−1.58i)T+(14.7+8.5i)T2 1 + (-5.92 - 1.58i)T + (14.7 + 8.5i)T^{2}
19 1+(−0.715−0.413i)T+(9.5+16.4i)T2 1 + (-0.715 - 0.413i)T + (9.5 + 16.4i)T^{2}
23 1+(−1.33+0.359i)T+(19.9−11.5i)T2 1 + (-1.33 + 0.359i)T + (19.9 - 11.5i)T^{2}
29 1+7.45T+29T2 1 + 7.45T + 29T^{2}
31 1+(2.97+5.15i)T+(−15.5+26.8i)T2 1 + (2.97 + 5.15i)T + (-15.5 + 26.8i)T^{2}
37 1+(−2.12+0.568i)T+(32.0−18.5i)T2 1 + (-2.12 + 0.568i)T + (32.0 - 18.5i)T^{2}
41 1+2.00iT−41T2 1 + 2.00iT - 41T^{2}
43 1+(1.73−1.73i)T−43iT2 1 + (1.73 - 1.73i)T - 43iT^{2}
47 1+(−1.11−4.17i)T+(−40.7+23.5i)T2 1 + (-1.11 - 4.17i)T + (-40.7 + 23.5i)T^{2}
53 1+(1.78−6.65i)T+(−45.8−26.5i)T2 1 + (1.78 - 6.65i)T + (-45.8 - 26.5i)T^{2}
59 1+(−4.07−7.06i)T+(−29.5+51.0i)T2 1 + (-4.07 - 7.06i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.12+1.94i)T+(−30.5−52.8i)T2 1 + (-1.12 + 1.94i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.25−4.67i)T+(−58.0−33.5i)T2 1 + (1.25 - 4.67i)T + (-58.0 - 33.5i)T^{2}
71 1−9.40iT−71T2 1 - 9.40iT - 71T^{2}
73 1+(−10.4−2.80i)T+(63.2+36.5i)T2 1 + (-10.4 - 2.80i)T + (63.2 + 36.5i)T^{2}
79 1+(7.72+4.46i)T+(39.5+68.4i)T2 1 + (7.72 + 4.46i)T + (39.5 + 68.4i)T^{2}
83 1+(2.36+2.36i)T+83iT2 1 + (2.36 + 2.36i)T + 83iT^{2}
89 1+(−4.69+8.13i)T+(−44.5−77.0i)T2 1 + (-4.69 + 8.13i)T + (-44.5 - 77.0i)T^{2}
97 1+(−5.63−5.63i)T+97iT2 1 + (-5.63 - 5.63i)T + 97iT^{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

−11.69008686304129940659150667987, −10.92912861893460060651355737737, −10.05108000626380391209442952853, −9.333553990578088463853026629865, −8.056322865336538469323845008582, −7.29608817055208952834753965238, −5.60911884778910876767163940662, −4.27093501290204521599878394672, −2.94804249808838000438936688527, −2.06203022159057065991045146852, 0.44334264377295833310907462448, 3.34769780875126347097863252459, 5.10309085277425152679668956550, 5.39349297853356615406132608961, 7.11460602741501979270377590427, 7.84708361764197707874015632326, 8.131232026403959303081481482155, 9.420363236211734240279961518092, 10.47246328247614792028918407329, 11.50603368754165879704479398349

Graph of the ZZ-function along the critical line