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Properties

Label 2-2695-2695.1649-c0-0-0
Degree 22
Conductor 26952695
Sign −0.972−0.232i-0.972 - 0.232i
Analytic cond. 1.344981.34498
Root an. cond. 1.159731.15973
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.147 − 1.97i)2-s + (−2.87 + 0.433i)4-s + (−0.733 + 0.680i)5-s + (−0.900 + 0.433i)7-s + (0.841 + 3.68i)8-s + (0.826 + 0.563i)9-s + (1.44 + 1.34i)10-s + (0.826 − 0.563i)11-s + (−1.48 − 0.716i)13-s + (0.988 + 1.71i)14-s + (4.36 − 1.34i)16-s + (−0.162 − 0.414i)17-s + (0.988 − 1.71i)18-s + (1.81 − 2.27i)20-s + (−1.23 − 1.54i)22-s + ⋯
L(s)  = 1  + (−0.147 − 1.97i)2-s + (−2.87 + 0.433i)4-s + (−0.733 + 0.680i)5-s + (−0.900 + 0.433i)7-s + (0.841 + 3.68i)8-s + (0.826 + 0.563i)9-s + (1.44 + 1.34i)10-s + (0.826 − 0.563i)11-s + (−1.48 − 0.716i)13-s + (0.988 + 1.71i)14-s + (4.36 − 1.34i)16-s + (−0.162 − 0.414i)17-s + (0.988 − 1.71i)18-s + (1.81 − 2.27i)20-s + (−1.23 − 1.54i)22-s + ⋯

Functional equation

Λ(s)=(2695s/2 ΓC(s) L(s)=((−0.972−0.232i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2695s/2 ΓC(s) L(s)=((−0.972−0.232i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26952695    =    5⋅72⋅115 \cdot 7^{2} \cdot 11
Sign: −0.972−0.232i-0.972 - 0.232i
Analytic conductor: 1.344981.34498
Root analytic conductor: 1.159731.15973
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2695(1649,⋅)\chi_{2695} (1649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2695, ( :0), −0.972−0.232i)(2,\ 2695,\ (\ :0),\ -0.972 - 0.232i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.49208525040.4920852504
L(12)L(\frac12) ≈\approx 0.49208525040.4920852504
L(1)L(1) 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)
bad5 1+(0.733−0.680i)T 1 + (0.733 - 0.680i)T
7 1+(0.900−0.433i)T 1 + (0.900 - 0.433i)T
11 1+(−0.826+0.563i)T 1 + (-0.826 + 0.563i)T
good2 1+(0.147+1.97i)T+(−0.988+0.149i)T2 1 + (0.147 + 1.97i)T + (-0.988 + 0.149i)T^{2}
3 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
13 1+(1.48+0.716i)T+(0.623+0.781i)T2 1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2}
17 1+(0.162+0.414i)T+(−0.733+0.680i)T2 1 + (0.162 + 0.414i)T + (-0.733 + 0.680i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(0.733+0.680i)T2 1 + (0.733 + 0.680i)T^{2}
29 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
31 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
41 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
43 1+(0.0332−0.145i)T+(−0.900−0.433i)T2 1 + (0.0332 - 0.145i)T + (-0.900 - 0.433i)T^{2}
47 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)T^{2}
53 1+(−0.955+0.294i)T2 1 + (-0.955 + 0.294i)T^{2}
59 1+(1.21+1.12i)T+(0.0747+0.997i)T2 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2}
61 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(1.23+1.54i)T+(−0.222+0.974i)T2 1 + (1.23 + 1.54i)T + (-0.222 + 0.974i)T^{2}
73 1+(−0.142+1.90i)T+(−0.988−0.149i)T2 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(−0.900+0.433i)T+(0.623−0.781i)T2 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2}
89 1+(−0.123−0.0841i)T+(0.365+0.930i)T2 1 + (-0.123 - 0.0841i)T + (0.365 + 0.930i)T^{2}
97 1−T2 1 - 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

−9.045609750445057255466904056179, −7.999394188313837331973243254214, −7.44802792055704216702156731220, −6.23759725052968443967798785808, −5.00757818747293625639934973915, −4.34582584295565405906368454952, −3.41970520245272223437057371787, −2.86989917044289946986208211331, −2.01400885439452585163364512414, −0.40728778331936876951914854232, 1.13411094084994508549246265985, 3.58485084488143939775264049597, 4.37572848012329689026584300911, 4.63547219861261467961533953481, 5.79529468553278497755368008840, 6.78286995904782219202965349288, 7.01540680618644955146335891347, 7.60643149833483143572621683215, 8.599992419272782682275751805787, 9.228439697233635975401487620532

Graph of the ZZ-function along the critical line