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Properties

Label 2-2175-5.4-c1-0-25
Degree 22
Conductor 21752175
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 17.367417.3674
Root an. cond. 4.167424.16742
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  − 1.98i·2-s − i·3-s − 1.93·4-s − 1.98·6-s + 2.16i·7-s − 0.119i·8-s − 9-s + 3.44·11-s + 1.93i·12-s + 3.74i·13-s + 4.29·14-s − 4.11·16-s + 4.33i·17-s + 1.98i·18-s − 3.90·19-s + ⋯
L(s)  = 1  − 1.40i·2-s − 0.577i·3-s − 0.969·4-s − 0.810·6-s + 0.818i·7-s − 0.0423i·8-s − 0.333·9-s + 1.03·11-s + 0.559i·12-s + 1.03i·13-s + 1.14·14-s − 1.02·16-s + 1.05i·17-s + 0.467i·18-s − 0.895·19-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+1/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 17.367417.3674
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2175(349,⋅)\chi_{2175} (349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2175, ( :1/2), 0.447+0.894i)(2,\ 2175,\ (\ :1/2),\ 0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.6563330951.656333095
L(12)L(\frac12) ≈\approx 1.6563330951.656333095
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+iT 1 + iT
5 1 1
29 1+T 1 + T
good2 1+1.98iT−2T2 1 + 1.98iT - 2T^{2}
7 1−2.16iT−7T2 1 - 2.16iT - 7T^{2}
11 1−3.44T+11T2 1 - 3.44T + 11T^{2}
13 1−3.74iT−13T2 1 - 3.74iT - 13T^{2}
17 1−4.33iT−17T2 1 - 4.33iT - 17T^{2}
19 1+3.90T+19T2 1 + 3.90T + 19T^{2}
23 1−3.78iT−23T2 1 - 3.78iT - 23T^{2}
31 1−10.3T+31T2 1 - 10.3T + 31T^{2}
37 1+7.70iT−37T2 1 + 7.70iT - 37T^{2}
41 1−7.88T+41T2 1 - 7.88T + 41T^{2}
43 1+0.975iT−43T2 1 + 0.975iT - 43T^{2}
47 1−12.1iT−47T2 1 - 12.1iT - 47T^{2}
53 1−13.1iT−53T2 1 - 13.1iT - 53T^{2}
59 1+1.47T+59T2 1 + 1.47T + 59T^{2}
61 1+4.24T+61T2 1 + 4.24T + 61T^{2}
67 1+6.74iT−67T2 1 + 6.74iT - 67T^{2}
71 1−10.3T+71T2 1 - 10.3T + 71T^{2}
73 1−12.3iT−73T2 1 - 12.3iT - 73T^{2}
79 1−5.02T+79T2 1 - 5.02T + 79T^{2}
83 1+10.4iT−83T2 1 + 10.4iT - 83T^{2}
89 1+4.35T+89T2 1 + 4.35T + 89T^{2}
97 1−3.75iT−97T2 1 - 3.75iT - 97T^{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

−9.146301521490862210793744339437, −8.488261443182238501037329919047, −7.41288768195546252697209858504, −6.39556158962876571017533101314, −5.98372758835787152325711404898, −4.46744238298049686312617556217, −3.91275194773925957129301406632, −2.74989407356915575498917317117, −1.98157822098564671085196866245, −1.17918097434105750771515152287, 0.65235584216096799006255581836, 2.56440043298082356769056289853, 3.78511780981422966365630727873, 4.61466438879545812686495472801, 5.23188119011665723039236717466, 6.37028954278872944143861292222, 6.66231662728529281964858147757, 7.64141465500375073726852332220, 8.301578776947915879886720936961, 8.939209343383113640063837920574

Graph of the ZZ-function along the critical line