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Properties

Label 2-315-105.2-c1-0-10
Degree 22
Conductor 315315
Sign 0.116+0.993i0.116 + 0.993i
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  + (−0.271 − 1.01i)2-s + (0.777 − 0.449i)4-s + (0.310 + 2.21i)5-s + (−1.26 − 2.32i)7-s + (−2.15 − 2.15i)8-s + (2.16 − 0.915i)10-s + (1.52 − 0.880i)11-s + (4.98 − 4.98i)13-s + (−2.00 + 1.91i)14-s + (−0.698 + 1.20i)16-s + (−0.458 − 0.122i)17-s + (2.33 + 1.34i)19-s + (1.23 + 1.58i)20-s + (−1.30 − 1.30i)22-s + (6.30 − 1.69i)23-s + ⋯
L(s)  = 1  + (−0.192 − 0.716i)2-s + (0.388 − 0.224i)4-s + (0.138 + 0.990i)5-s + (−0.479 − 0.877i)7-s + (−0.760 − 0.760i)8-s + (0.683 − 0.289i)10-s + (0.460 − 0.265i)11-s + (1.38 − 1.38i)13-s + (−0.537 + 0.512i)14-s + (−0.174 + 0.302i)16-s + (−0.111 − 0.0298i)17-s + (0.535 + 0.309i)19-s + (0.276 + 0.354i)20-s + (−0.278 − 0.278i)22-s + (1.31 − 0.352i)23-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=((0.116+0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+1/2) L(s)=((0.116+0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: 0.116+0.993i0.116 + 0.993i
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), 0.116+0.993i)(2,\ 315,\ (\ :1/2),\ 0.116 + 0.993i)

Particular Values

L(1)L(1) ≈\approx 1.00236−0.891652i1.00236 - 0.891652i
L(12)L(\frac12) ≈\approx 1.00236−0.891652i1.00236 - 0.891652i
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+(−0.310−2.21i)T 1 + (-0.310 - 2.21i)T
7 1+(1.26+2.32i)T 1 + (1.26 + 2.32i)T
good2 1+(0.271+1.01i)T+(−1.73+i)T2 1 + (0.271 + 1.01i)T + (-1.73 + i)T^{2}
11 1+(−1.52+0.880i)T+(5.5−9.52i)T2 1 + (-1.52 + 0.880i)T + (5.5 - 9.52i)T^{2}
13 1+(−4.98+4.98i)T−13iT2 1 + (-4.98 + 4.98i)T - 13iT^{2}
17 1+(0.458+0.122i)T+(14.7+8.5i)T2 1 + (0.458 + 0.122i)T + (14.7 + 8.5i)T^{2}
19 1+(−2.33−1.34i)T+(9.5+16.4i)T2 1 + (-2.33 - 1.34i)T + (9.5 + 16.4i)T^{2}
23 1+(−6.30+1.69i)T+(19.9−11.5i)T2 1 + (-6.30 + 1.69i)T + (19.9 - 11.5i)T^{2}
29 1+7.01T+29T2 1 + 7.01T + 29T^{2}
31 1+(−3.95−6.84i)T+(−15.5+26.8i)T2 1 + (-3.95 - 6.84i)T + (-15.5 + 26.8i)T^{2}
37 1+(6.45−1.73i)T+(32.0−18.5i)T2 1 + (6.45 - 1.73i)T + (32.0 - 18.5i)T^{2}
41 1+6.20iT−41T2 1 + 6.20iT - 41T^{2}
43 1+(2.28−2.28i)T−43iT2 1 + (2.28 - 2.28i)T - 43iT^{2}
47 1+(−0.393−1.47i)T+(−40.7+23.5i)T2 1 + (-0.393 - 1.47i)T + (-40.7 + 23.5i)T^{2}
53 1+(2.26−8.45i)T+(−45.8−26.5i)T2 1 + (2.26 - 8.45i)T + (-45.8 - 26.5i)T^{2}
59 1+(−6.17−10.7i)T+(−29.5+51.0i)T2 1 + (-6.17 - 10.7i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.26−3.91i)T+(−30.5−52.8i)T2 1 + (2.26 - 3.91i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.530−1.98i)T+(−58.0−33.5i)T2 1 + (0.530 - 1.98i)T + (-58.0 - 33.5i)T^{2}
71 1+9.79iT−71T2 1 + 9.79iT - 71T^{2}
73 1+(1.45+0.391i)T+(63.2+36.5i)T2 1 + (1.45 + 0.391i)T + (63.2 + 36.5i)T^{2}
79 1+(−9.08−5.24i)T+(39.5+68.4i)T2 1 + (-9.08 - 5.24i)T + (39.5 + 68.4i)T^{2}
83 1+(−2.07−2.07i)T+83iT2 1 + (-2.07 - 2.07i)T + 83iT^{2}
89 1+(1.64−2.85i)T+(−44.5−77.0i)T2 1 + (1.64 - 2.85i)T + (-44.5 - 77.0i)T^{2}
97 1+(−3.21−3.21i)T+97iT2 1 + (-3.21 - 3.21i)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

−10.98891973598577154168377713234, −10.73964761436443235769481057262, −9.979168038287854244614818926243, −8.850831785982318317912503265358, −7.40935926710054330592941424063, −6.60532091253162385557262733208, −5.71874581594749194568515617885, −3.60122329924608024647323641637, −3.04483942163514059628854231155, −1.16628487039200935812765593635, 1.89597197117038354513015934715, 3.61657265759554096277690878136, 5.14471999768315894877327339257, 6.14572409182681565000113345903, 6.90261446601307368916664607749, 8.206703024199377338571576118864, 9.020849616006131474583868020706, 9.452856773281140720889864223872, 11.37609188272756020703578260650, 11.69174642420970983274282278623

Graph of the ZZ-function along the critical line