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Properties

Label 2-352-88.69-c1-0-9
Degree 22
Conductor 352352
Sign 0.829+0.559i0.829 + 0.559i
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  + (2.62 − 0.852i)3-s + (1.70 − 2.35i)5-s + (−0.730 + 2.24i)7-s + (3.72 − 2.70i)9-s + (−2.02 + 2.62i)11-s + (1.49 + 2.05i)13-s + (2.47 − 7.62i)15-s + (−4.77 − 3.46i)17-s + (0.252 − 0.0821i)19-s + 6.51i·21-s − 4.18·23-s + (−1.06 − 3.27i)25-s + (2.60 − 3.58i)27-s + (−3.08 − 1.00i)29-s + (1.66 − 1.20i)31-s + ⋯
L(s)  = 1  + (1.51 − 0.492i)3-s + (0.763 − 1.05i)5-s + (−0.276 + 0.849i)7-s + (1.24 − 0.902i)9-s + (−0.609 + 0.792i)11-s + (0.414 + 0.571i)13-s + (0.639 − 1.96i)15-s + (−1.15 − 0.841i)17-s + (0.0580 − 0.0188i)19-s + 1.42i·21-s − 0.871·23-s + (−0.212 − 0.654i)25-s + (0.500 − 0.689i)27-s + (−0.573 − 0.186i)29-s + (0.298 − 0.217i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.829+0.559i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.829+0.559i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.829+0.559i0.829 + 0.559i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(113,⋅)\chi_{352} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.829+0.559i)(2,\ 352,\ (\ :1/2),\ 0.829 + 0.559i)

Particular Values

L(1)L(1) ≈\approx 2.12049−0.648326i2.12049 - 0.648326i
L(12)L(\frac12) ≈\approx 2.12049−0.648326i2.12049 - 0.648326i
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+(2.02−2.62i)T 1 + (2.02 - 2.62i)T
good3 1+(−2.62+0.852i)T+(2.42−1.76i)T2 1 + (-2.62 + 0.852i)T + (2.42 - 1.76i)T^{2}
5 1+(−1.70+2.35i)T+(−1.54−4.75i)T2 1 + (-1.70 + 2.35i)T + (-1.54 - 4.75i)T^{2}
7 1+(0.730−2.24i)T+(−5.66−4.11i)T2 1 + (0.730 - 2.24i)T + (-5.66 - 4.11i)T^{2}
13 1+(−1.49−2.05i)T+(−4.01+12.3i)T2 1 + (-1.49 - 2.05i)T + (-4.01 + 12.3i)T^{2}
17 1+(4.77+3.46i)T+(5.25+16.1i)T2 1 + (4.77 + 3.46i)T + (5.25 + 16.1i)T^{2}
19 1+(−0.252+0.0821i)T+(15.3−11.1i)T2 1 + (-0.252 + 0.0821i)T + (15.3 - 11.1i)T^{2}
23 1+4.18T+23T2 1 + 4.18T + 23T^{2}
29 1+(3.08+1.00i)T+(23.4+17.0i)T2 1 + (3.08 + 1.00i)T + (23.4 + 17.0i)T^{2}
31 1+(−1.66+1.20i)T+(9.57−29.4i)T2 1 + (-1.66 + 1.20i)T + (9.57 - 29.4i)T^{2}
37 1+(−0.740−0.240i)T+(29.9+21.7i)T2 1 + (-0.740 - 0.240i)T + (29.9 + 21.7i)T^{2}
41 1+(−0.927−2.85i)T+(−33.1+24.0i)T2 1 + (-0.927 - 2.85i)T + (-33.1 + 24.0i)T^{2}
43 1−1.14iT−43T2 1 - 1.14iT - 43T^{2}
47 1+(2.98+9.17i)T+(−38.0+27.6i)T2 1 + (2.98 + 9.17i)T + (-38.0 + 27.6i)T^{2}
53 1+(−5.77−7.94i)T+(−16.3+50.4i)T2 1 + (-5.77 - 7.94i)T + (-16.3 + 50.4i)T^{2}
59 1+(−5.50−1.78i)T+(47.7+34.6i)T2 1 + (-5.50 - 1.78i)T + (47.7 + 34.6i)T^{2}
61 1+(−3.60+4.96i)T+(−18.8−58.0i)T2 1 + (-3.60 + 4.96i)T + (-18.8 - 58.0i)T^{2}
67 1+0.241iT−67T2 1 + 0.241iT - 67T^{2}
71 1+(3.19+2.31i)T+(21.9+67.5i)T2 1 + (3.19 + 2.31i)T + (21.9 + 67.5i)T^{2}
73 1+(1.96−6.04i)T+(−59.0−42.9i)T2 1 + (1.96 - 6.04i)T + (-59.0 - 42.9i)T^{2}
79 1+(4.19−3.04i)T+(24.4−75.1i)T2 1 + (4.19 - 3.04i)T + (24.4 - 75.1i)T^{2}
83 1+(−8.95+12.3i)T+(−25.6−78.9i)T2 1 + (-8.95 + 12.3i)T + (-25.6 - 78.9i)T^{2}
89 1+16.2T+89T2 1 + 16.2T + 89T^{2}
97 1+(−5.90+4.29i)T+(29.9−92.2i)T2 1 + (-5.90 + 4.29i)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

−11.66739284801292902470377847160, −9.994814290080342838487945727562, −9.201954348455272815367697659254, −8.823008674356638428614657349858, −7.86212528462603334593609489034, −6.74383944630310251337112664784, −5.47047049541208169219723206118, −4.25394767386274719076506209493, −2.59598944853824244675385087426, −1.85708300933882109647483822960, 2.23288382344851194260233454700, 3.22349253358061459609767067357, 4.07794101290353225831727484526, 5.84918372282719075067619729096, 6.91006121730906886114712630420, 7.993019095499844344006061594710, 8.738078400104865773526066457070, 9.870627845106617380209994634700, 10.42050634650237511566516760911, 11.08322561976670536056240802510

Graph of the ZZ-function along the critical line