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Properties

Label 2-352-44.39-c1-0-9
Degree 22
Conductor 352352
Sign 0.371+0.928i0.371 + 0.928i
Analytic cond. 2.810732.81073
Root an. cond. 1.676521.67652
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  + (0.708 − 0.975i)3-s + (0.671 − 2.06i)5-s + (1.18 − 0.858i)7-s + (0.477 + 1.47i)9-s + (−3.29 − 0.371i)11-s + (5.50 − 1.78i)13-s + (−1.54 − 2.11i)15-s + (−2.83 − 0.921i)17-s + (−4.63 − 3.36i)19-s − 1.76i·21-s − 0.400i·23-s + (0.222 + 0.162i)25-s + (5.21 + 1.69i)27-s + (2.26 + 3.11i)29-s + (−1.41 + 0.460i)31-s + ⋯
L(s)  = 1  + (0.409 − 0.563i)3-s + (0.300 − 0.924i)5-s + (0.446 − 0.324i)7-s + (0.159 + 0.490i)9-s + (−0.993 − 0.112i)11-s + (1.52 − 0.496i)13-s + (−0.397 − 0.547i)15-s + (−0.688 − 0.223i)17-s + (−1.06 − 0.773i)19-s − 0.384i·21-s − 0.0834i·23-s + (0.0445 + 0.0324i)25-s + (1.00 + 0.325i)27-s + (0.419 + 0.577i)29-s + (−0.254 + 0.0827i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.371+0.928i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.371+0.928i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.371+0.928i0.371 + 0.928i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(127,⋅)\chi_{352} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.371+0.928i)(2,\ 352,\ (\ :1/2),\ 0.371 + 0.928i)

Particular Values

L(1)L(1) ≈\approx 1.34495−0.910221i1.34495 - 0.910221i
L(12)L(\frac12) ≈\approx 1.34495−0.910221i1.34495 - 0.910221i
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
11 1+(3.29+0.371i)T 1 + (3.29 + 0.371i)T
good3 1+(−0.708+0.975i)T+(−0.927−2.85i)T2 1 + (-0.708 + 0.975i)T + (-0.927 - 2.85i)T^{2}
5 1+(−0.671+2.06i)T+(−4.04−2.93i)T2 1 + (-0.671 + 2.06i)T + (-4.04 - 2.93i)T^{2}
7 1+(−1.18+0.858i)T+(2.16−6.65i)T2 1 + (-1.18 + 0.858i)T + (2.16 - 6.65i)T^{2}
13 1+(−5.50+1.78i)T+(10.5−7.64i)T2 1 + (-5.50 + 1.78i)T + (10.5 - 7.64i)T^{2}
17 1+(2.83+0.921i)T+(13.7+9.99i)T2 1 + (2.83 + 0.921i)T + (13.7 + 9.99i)T^{2}
19 1+(4.63+3.36i)T+(5.87+18.0i)T2 1 + (4.63 + 3.36i)T + (5.87 + 18.0i)T^{2}
23 1+0.400iT−23T2 1 + 0.400iT - 23T^{2}
29 1+(−2.26−3.11i)T+(−8.96+27.5i)T2 1 + (-2.26 - 3.11i)T + (-8.96 + 27.5i)T^{2}
31 1+(1.41−0.460i)T+(25.0−18.2i)T2 1 + (1.41 - 0.460i)T + (25.0 - 18.2i)T^{2}
37 1+(−5.01+3.64i)T+(11.4−35.1i)T2 1 + (-5.01 + 3.64i)T + (11.4 - 35.1i)T^{2}
41 1+(2.50−3.45i)T+(−12.6−38.9i)T2 1 + (2.50 - 3.45i)T + (-12.6 - 38.9i)T^{2}
43 1−6.17T+43T2 1 - 6.17T + 43T^{2}
47 1+(4.79−6.59i)T+(−14.5−44.6i)T2 1 + (4.79 - 6.59i)T + (-14.5 - 44.6i)T^{2}
53 1+(0.633+1.95i)T+(−42.8+31.1i)T2 1 + (0.633 + 1.95i)T + (-42.8 + 31.1i)T^{2}
59 1+(−7.19−9.90i)T+(−18.2+56.1i)T2 1 + (-7.19 - 9.90i)T + (-18.2 + 56.1i)T^{2}
61 1+(12.4+4.04i)T+(49.3+35.8i)T2 1 + (12.4 + 4.04i)T + (49.3 + 35.8i)T^{2}
67 1−2.72iT−67T2 1 - 2.72iT - 67T^{2}
71 1+(−9.62−3.12i)T+(57.4+41.7i)T2 1 + (-9.62 - 3.12i)T + (57.4 + 41.7i)T^{2}
73 1+(−2.77−3.81i)T+(−22.5+69.4i)T2 1 + (-2.77 - 3.81i)T + (-22.5 + 69.4i)T^{2}
79 1+(2.34+7.20i)T+(−63.9+46.4i)T2 1 + (2.34 + 7.20i)T + (-63.9 + 46.4i)T^{2}
83 1+(3.31−10.1i)T+(−67.1−48.7i)T2 1 + (3.31 - 10.1i)T + (-67.1 - 48.7i)T^{2}
89 1−1.16T+89T2 1 - 1.16T + 89T^{2}
97 1+(−5.15−15.8i)T+(−78.4+57.0i)T2 1 + (-5.15 - 15.8i)T + (-78.4 + 57.0i)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

−10.99091276572895667942904519356, −10.71159209462170840084638606419, −9.161180032827287290618901254768, −8.409261940182769167529954320380, −7.78656902353057562155756131017, −6.54005063987887360482569394160, −5.30142271177874437391274195129, −4.38387248517427633075870587551, −2.63099320147368262214905205406, −1.24114898896623376475622490864, 2.13141761680469463215722684236, 3.42708608729147455232122641181, 4.47292318058643453576010630414, 5.98822870669951759100926882329, 6.70001748084125243998993024282, 8.151769769386247179846771222762, 8.804037303296103468998653629905, 9.956587215132374124696924203812, 10.66199751319597310528108271423, 11.36732863451861794502341115790

Graph of the ZZ-function along the critical line