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Properties

Label 2-475-19.7-c1-0-12
Degree 22
Conductor 475475
Sign −0.981−0.189i-0.981 - 0.189i
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
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  + (1.08 + 1.87i)2-s + (1.47 + 2.55i)3-s + (−1.35 + 2.34i)4-s + (−3.20 + 5.54i)6-s − 0.591·7-s − 1.53·8-s + (−2.85 + 4.94i)9-s + 2.58·11-s − 7.99·12-s + (3.43 − 5.94i)13-s + (−0.641 − 1.11i)14-s + (1.03 + 1.79i)16-s + (−2.61 − 4.53i)17-s − 12.3·18-s + (−2.26 − 3.72i)19-s + ⋯
L(s)  = 1  + (0.767 + 1.32i)2-s + (0.852 + 1.47i)3-s + (−0.677 + 1.17i)4-s + (−1.30 + 2.26i)6-s − 0.223·7-s − 0.544·8-s + (−0.952 + 1.64i)9-s + 0.778·11-s − 2.30·12-s + (0.952 − 1.64i)13-s + (−0.171 − 0.297i)14-s + (0.259 + 0.449i)16-s + (−0.634 − 1.09i)17-s − 2.92·18-s + (−0.519 − 0.854i)19-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((−0.981−0.189i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((−0.981−0.189i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: −0.981−0.189i-0.981 - 0.189i
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ475(26,⋅)\chi_{475} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 475, ( :1/2), −0.981−0.189i)(2,\ 475,\ (\ :1/2),\ -0.981 - 0.189i)

Particular Values

L(1)L(1) ≈\approx 0.256794+2.69146i0.256794 + 2.69146i
L(12)L(\frac12) ≈\approx 0.256794+2.69146i0.256794 + 2.69146i
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)
bad5 1 1
19 1+(2.26+3.72i)T 1 + (2.26 + 3.72i)T
good2 1+(−1.08−1.87i)T+(−1+1.73i)T2 1 + (-1.08 - 1.87i)T + (-1 + 1.73i)T^{2}
3 1+(−1.47−2.55i)T+(−1.5+2.59i)T2 1 + (-1.47 - 2.55i)T + (-1.5 + 2.59i)T^{2}
7 1+0.591T+7T2 1 + 0.591T + 7T^{2}
11 1−2.58T+11T2 1 - 2.58T + 11T^{2}
13 1+(−3.43+5.94i)T+(−6.5−11.2i)T2 1 + (-3.43 + 5.94i)T + (-6.5 - 11.2i)T^{2}
17 1+(2.61+4.53i)T+(−8.5+14.7i)T2 1 + (2.61 + 4.53i)T + (-8.5 + 14.7i)T^{2}
23 1+(−1.45+2.51i)T+(−11.5−19.9i)T2 1 + (-1.45 + 2.51i)T + (-11.5 - 19.9i)T^{2}
29 1+(3.52−6.10i)T+(−14.5−25.1i)T2 1 + (3.52 - 6.10i)T + (-14.5 - 25.1i)T^{2}
31 1+6.81T+31T2 1 + 6.81T + 31T^{2}
37 1−4.82T+37T2 1 - 4.82T + 37T^{2}
41 1+(3.11+5.39i)T+(−20.5+35.5i)T2 1 + (3.11 + 5.39i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2.18−3.77i)T+(−21.5+37.2i)T2 1 + (-2.18 - 3.77i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.27+2.21i)T+(−23.5−40.7i)T2 1 + (-1.27 + 2.21i)T + (-23.5 - 40.7i)T^{2}
53 1+(4.79−8.30i)T+(−26.5−45.8i)T2 1 + (4.79 - 8.30i)T + (-26.5 - 45.8i)T^{2}
59 1+(−1.46−2.53i)T+(−29.5+51.0i)T2 1 + (-1.46 - 2.53i)T + (-29.5 + 51.0i)T^{2}
61 1+(1.16−2.01i)T+(−30.5−52.8i)T2 1 + (1.16 - 2.01i)T + (-30.5 - 52.8i)T^{2}
67 1+(−2.15+3.72i)T+(−33.5−58.0i)T2 1 + (-2.15 + 3.72i)T + (-33.5 - 58.0i)T^{2}
71 1+(−6.74−11.6i)T+(−35.5+61.4i)T2 1 + (-6.74 - 11.6i)T + (-35.5 + 61.4i)T^{2}
73 1+(4.21+7.29i)T+(−36.5+63.2i)T2 1 + (4.21 + 7.29i)T + (-36.5 + 63.2i)T^{2}
79 1+(2.93+5.08i)T+(−39.5+68.4i)T2 1 + (2.93 + 5.08i)T + (-39.5 + 68.4i)T^{2}
83 1−4.02T+83T2 1 - 4.02T + 83T^{2}
89 1+(1.85−3.21i)T+(−44.5−77.0i)T2 1 + (1.85 - 3.21i)T + (-44.5 - 77.0i)T^{2}
97 1+(−1.26−2.18i)T+(−48.5+84.0i)T2 1 + (-1.26 - 2.18i)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

−11.09246963696288882341309426112, −10.53197525102655686383753198835, −9.237301086008468488991451618615, −8.781876801359803646136799270714, −7.79354566619000198715250856038, −6.73307789951198154064492312507, −5.59115672517596570753913257860, −4.79272293951310965207754900199, −3.87086011278252867237573244936, −3.01032154388792757433379296117, 1.55886128742194172565913182763, 2.01877661776619515883369938187, 3.51782505641970141088888726531, 4.13983077620309079161761631065, 6.05928327172089374529583630387, 6.74232659714767712862559013074, 7.939430841861755470862365578477, 8.897005524381919266349680503928, 9.670808968296713867517520471557, 11.14407586293573320507040789856

Graph of the ZZ-function along the critical line