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Properties

Label 2-1183-13.12-c1-0-70
Degree 22
Conductor 11831183
Sign 0.722+0.691i0.722 + 0.691i
Analytic cond. 9.446309.44630
Root an. cond. 3.073483.07348
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  + 0.149i·2-s + 2.76·3-s + 1.97·4-s − 4.13i·5-s + 0.413i·6-s + i·7-s + 0.595i·8-s + 4.61·9-s + 0.618·10-s − 2.55i·11-s + 5.45·12-s − 0.149·14-s − 11.4i·15-s + 3.86·16-s − 1.50·17-s + 0.691i·18-s + ⋯
L(s)  = 1  + 0.105i·2-s + 1.59·3-s + 0.988·4-s − 1.84i·5-s + 0.168i·6-s + 0.377i·7-s + 0.210i·8-s + 1.53·9-s + 0.195·10-s − 0.769i·11-s + 1.57·12-s − 0.0399·14-s − 2.94i·15-s + 0.966·16-s − 0.364·17-s + 0.162i·18-s + ⋯

Functional equation

Λ(s)=(1183s/2 ΓC(s) L(s)=((0.722+0.691i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1183s/2 ΓC(s+1/2) L(s)=((0.722+0.691i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11831183    =    7⋅1327 \cdot 13^{2}
Sign: 0.722+0.691i0.722 + 0.691i
Analytic conductor: 9.446309.44630
Root analytic conductor: 3.073483.07348
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1183(337,⋅)\chi_{1183} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1183, ( :1/2), 0.722+0.691i)(2,\ 1183,\ (\ :1/2),\ 0.722 + 0.691i)

Particular Values

L(1)L(1) ≈\approx 3.4796066153.479606615
L(12)L(\frac12) ≈\approx 3.4796066153.479606615
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)
bad7 1−iT 1 - iT
13 1 1
good2 1−0.149iT−2T2 1 - 0.149iT - 2T^{2}
3 1−2.76T+3T2 1 - 2.76T + 3T^{2}
5 1+4.13iT−5T2 1 + 4.13iT - 5T^{2}
11 1+2.55iT−11T2 1 + 2.55iT - 11T^{2}
17 1+1.50T+17T2 1 + 1.50T + 17T^{2}
19 1−5.93iT−19T2 1 - 5.93iT - 19T^{2}
23 1+6.55T+23T2 1 + 6.55T + 23T^{2}
29 1−0.283T+29T2 1 - 0.283T + 29T^{2}
31 1−1.95iT−31T2 1 - 1.95iT - 31T^{2}
37 1−5.66iT−37T2 1 - 5.66iT - 37T^{2}
41 1−6.70iT−41T2 1 - 6.70iT - 41T^{2}
43 1−8.14T+43T2 1 - 8.14T + 43T^{2}
47 1+3.94iT−47T2 1 + 3.94iT - 47T^{2}
53 1+1.08T+53T2 1 + 1.08T + 53T^{2}
59 1+3.71iT−59T2 1 + 3.71iT - 59T^{2}
61 1−1.93T+61T2 1 - 1.93T + 61T^{2}
67 1−3.38iT−67T2 1 - 3.38iT - 67T^{2}
71 1+5.36iT−71T2 1 + 5.36iT - 71T^{2}
73 1−2.62iT−73T2 1 - 2.62iT - 73T^{2}
79 1−7.89T+79T2 1 - 7.89T + 79T^{2}
83 1−10.5iT−83T2 1 - 10.5iT - 83T^{2}
89 1+6.64iT−89T2 1 + 6.64iT - 89T^{2}
97 1+0.504iT−97T2 1 + 0.504iT - 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

−9.486734374020988338081569451638, −8.544124550528839889558587744585, −8.249956034569618938604169514227, −7.65948581388541983394153398303, −6.25887107245756455092538778699, −5.48321798323146671787709539949, −4.25930735406136291027524222629, −3.38198863691151545591524597367, −2.20147294258284861136431629560, −1.38802875215899210107331621423, 2.15841324931164942804046852482, 2.44500561719935921308275660884, 3.39945439270224790847016242865, 4.15504092407431979811326667598, 6.03910535040330605852399947731, 6.92727048988158527465841550300, 7.36666081774686506000211748919, 7.912147658294991219985834108674, 9.177223084706835188647483102393, 9.926712548473340648731981782464

Graph of the ZZ-function along the critical line