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Properties

Label 2-91-13.4-c1-0-2
Degree 22
Conductor 9191
Sign 0.313−0.949i0.313 - 0.949i
Analytic cond. 0.7266380.726638
Root an. cond. 0.8524310.852431
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  + (1.82 + 1.05i)2-s + (−1.13 + 1.95i)3-s + (1.22 + 2.12i)4-s − 3.60i·5-s + (−4.13 + 2.38i)6-s + (−0.866 + 0.5i)7-s + 0.948i·8-s + (−1.05 − 1.83i)9-s + (3.79 − 6.57i)10-s + (0.767 + 0.443i)11-s − 5.53·12-s + (−1.17 + 3.40i)13-s − 2.10·14-s + (7.05 + 4.07i)15-s + (1.44 − 2.51i)16-s + (−2.48 − 4.29i)17-s + ⋯
L(s)  = 1  + (1.29 + 0.745i)2-s + (−0.652 + 1.13i)3-s + (0.612 + 1.06i)4-s − 1.61i·5-s + (−1.68 + 0.973i)6-s + (−0.327 + 0.188i)7-s + 0.335i·8-s + (−0.352 − 0.610i)9-s + (1.20 − 2.08i)10-s + (0.231 + 0.133i)11-s − 1.59·12-s + (−0.325 + 0.945i)13-s − 0.563·14-s + (1.82 + 1.05i)15-s + (0.362 − 0.627i)16-s + (−0.601 − 1.04i)17-s + ⋯

Functional equation

Λ(s)=(91s/2 ΓC(s) L(s)=((0.313−0.949i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(91s/2 ΓC(s+1/2) L(s)=((0.313−0.949i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9191    =    7⋅137 \cdot 13
Sign: 0.313−0.949i0.313 - 0.949i
Analytic conductor: 0.7266380.726638
Root analytic conductor: 0.8524310.852431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ91(43,⋅)\chi_{91} (43, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 91, ( :1/2), 0.313−0.949i)(2,\ 91,\ (\ :1/2),\ 0.313 - 0.949i)

Particular Values

L(1)L(1) ≈\approx 1.18038+0.853527i1.18038 + 0.853527i
L(12)L(\frac12) ≈\approx 1.18038+0.853527i1.18038 + 0.853527i
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)
bad7 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
13 1+(1.17−3.40i)T 1 + (1.17 - 3.40i)T
good2 1+(−1.82−1.05i)T+(1+1.73i)T2 1 + (-1.82 - 1.05i)T + (1 + 1.73i)T^{2}
3 1+(1.13−1.95i)T+(−1.5−2.59i)T2 1 + (1.13 - 1.95i)T + (-1.5 - 2.59i)T^{2}
5 1+3.60iT−5T2 1 + 3.60iT - 5T^{2}
11 1+(−0.767−0.443i)T+(5.5+9.52i)T2 1 + (-0.767 - 0.443i)T + (5.5 + 9.52i)T^{2}
17 1+(2.48+4.29i)T+(−8.5+14.7i)T2 1 + (2.48 + 4.29i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.06+1.18i)T+(9.5−16.4i)T2 1 + (-2.06 + 1.18i)T + (9.5 - 16.4i)T^{2}
23 1+(1.92−3.34i)T+(−11.5−19.9i)T2 1 + (1.92 - 3.34i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.640−1.11i)T+(−14.5−25.1i)T2 1 + (0.640 - 1.11i)T + (-14.5 - 25.1i)T^{2}
31 1−8.46iT−31T2 1 - 8.46iT - 31T^{2}
37 1+(8.34+4.81i)T+(18.5+32.0i)T2 1 + (8.34 + 4.81i)T + (18.5 + 32.0i)T^{2}
41 1+(−10.4−6.04i)T+(20.5+35.5i)T2 1 + (-10.4 - 6.04i)T + (20.5 + 35.5i)T^{2}
43 1+(1.82+3.15i)T+(−21.5+37.2i)T2 1 + (1.82 + 3.15i)T + (-21.5 + 37.2i)T^{2}
47 1+2.98iT−47T2 1 + 2.98iT - 47T^{2}
53 1−4.92T+53T2 1 - 4.92T + 53T^{2}
59 1+(−6.34+3.66i)T+(29.5−51.0i)T2 1 + (-6.34 + 3.66i)T + (29.5 - 51.0i)T^{2}
61 1+(−0.769−1.33i)T+(−30.5+52.8i)T2 1 + (-0.769 - 1.33i)T + (-30.5 + 52.8i)T^{2}
67 1+(−7.29−4.21i)T+(33.5+58.0i)T2 1 + (-7.29 - 4.21i)T + (33.5 + 58.0i)T^{2}
71 1+(5.58−3.22i)T+(35.5−61.4i)T2 1 + (5.58 - 3.22i)T + (35.5 - 61.4i)T^{2}
73 1−7.14iT−73T2 1 - 7.14iT - 73T^{2}
79 1−0.757T+79T2 1 - 0.757T + 79T^{2}
83 1+4.76iT−83T2 1 + 4.76iT - 83T^{2}
89 1+(−3.13−1.80i)T+(44.5+77.0i)T2 1 + (-3.13 - 1.80i)T + (44.5 + 77.0i)T^{2}
97 1+(0.401−0.231i)T+(48.5−84.0i)T2 1 + (0.401 - 0.231i)T + (48.5 - 84.0i)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

−14.30245090119271805595220182504, −13.34785658818790305941839831742, −12.32207270096983413158476659352, −11.55513022146552498576152487929, −9.765348406127732885237439298572, −8.968856041429115878976106315126, −7.04840669033812002321486487885, −5.50068031658844402984369694698, −4.89957598556280342503211753749, −3.99794790832158864949485194602, 2.37281809325816280058643494239, 3.74834103153718575203641713323, 5.82471067815063300170082587770, 6.53703218661471713830901336496, 7.74469528359523336862338933431, 10.26392173641458628449549033867, 11.03144231686742927919000163645, 11.93724755004018094311959272545, 12.81055390391385226140366309789, 13.61120391086656579978546598347

Graph of the ZZ-function along the critical line