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Properties

Label 2-34e2-1156.747-c0-0-0
Degree 22
Conductor 11561156
Sign −0.887+0.460i-0.887 + 0.460i
Analytic cond. 0.5769190.576919
Root an. cond. 0.7595510.759551
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  + (0.850 − 0.526i)2-s + (0.445 − 0.895i)4-s + (−1.98 − 0.183i)5-s + (−0.0922 − 0.995i)8-s + (−0.932 + 0.361i)9-s + (−1.78 + 0.887i)10-s + (−0.172 − 1.85i)13-s + (−0.602 − 0.798i)16-s + (−0.602 − 0.798i)17-s + (−0.602 + 0.798i)18-s + (−1.04 + 1.69i)20-s + (2.91 + 0.544i)25-s + (−1.12 − 1.48i)26-s + (−0.646 − 0.322i)29-s + (−0.932 − 0.361i)32-s + ⋯
L(s)  = 1  + (0.850 − 0.526i)2-s + (0.445 − 0.895i)4-s + (−1.98 − 0.183i)5-s + (−0.0922 − 0.995i)8-s + (−0.932 + 0.361i)9-s + (−1.78 + 0.887i)10-s + (−0.172 − 1.85i)13-s + (−0.602 − 0.798i)16-s + (−0.602 − 0.798i)17-s + (−0.602 + 0.798i)18-s + (−1.04 + 1.69i)20-s + (2.91 + 0.544i)25-s + (−1.12 − 1.48i)26-s + (−0.646 − 0.322i)29-s + (−0.932 − 0.361i)32-s + ⋯

Functional equation

Λ(s)=(1156s/2 ΓC(s) L(s)=((−0.887+0.460i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1156s/2 ΓC(s) L(s)=((−0.887+0.460i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11561156    =    22⋅1722^{2} \cdot 17^{2}
Sign: −0.887+0.460i-0.887 + 0.460i
Analytic conductor: 0.5769190.576919
Root analytic conductor: 0.7595510.759551
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1156(747,⋅)\chi_{1156} (747, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1156, ( :0), −0.887+0.460i)(2,\ 1156,\ (\ :0),\ -0.887 + 0.460i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.83788465960.8378846596
L(12)L(\frac12) ≈\approx 0.83788465960.8378846596
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)
bad2 1+(−0.850+0.526i)T 1 + (-0.850 + 0.526i)T
17 1+(0.602+0.798i)T 1 + (0.602 + 0.798i)T
good3 1+(0.932−0.361i)T2 1 + (0.932 - 0.361i)T^{2}
5 1+(1.98+0.183i)T+(0.982+0.183i)T2 1 + (1.98 + 0.183i)T + (0.982 + 0.183i)T^{2}
7 1+(0.739−0.673i)T2 1 + (0.739 - 0.673i)T^{2}
11 1+(−0.602−0.798i)T2 1 + (-0.602 - 0.798i)T^{2}
13 1+(0.172+1.85i)T+(−0.982+0.183i)T2 1 + (0.172 + 1.85i)T + (-0.982 + 0.183i)T^{2}
19 1+(−0.445−0.895i)T2 1 + (-0.445 - 0.895i)T^{2}
23 1+(0.739−0.673i)T2 1 + (0.739 - 0.673i)T^{2}
29 1+(0.646+0.322i)T+(0.602+0.798i)T2 1 + (0.646 + 0.322i)T + (0.602 + 0.798i)T^{2}
31 1+(−0.982+0.183i)T2 1 + (-0.982 + 0.183i)T^{2}
37 1+(−1.01+0.288i)T+(0.850−0.526i)T2 1 + (-1.01 + 0.288i)T + (0.850 - 0.526i)T^{2}
41 1+(−0.365−1.95i)T+(−0.932+0.361i)T2 1 + (-0.365 - 1.95i)T + (-0.932 + 0.361i)T^{2}
43 1+(0.273+0.961i)T2 1 + (0.273 + 0.961i)T^{2}
47 1+(−0.739−0.673i)T2 1 + (-0.739 - 0.673i)T^{2}
53 1+(−1.37+0.533i)T+(0.739−0.673i)T2 1 + (-1.37 + 0.533i)T + (0.739 - 0.673i)T^{2}
59 1+(−0.0922−0.995i)T2 1 + (-0.0922 - 0.995i)T^{2}
61 1+(0.247+0.271i)T+(−0.0922+0.995i)T2 1 + (0.247 + 0.271i)T + (-0.0922 + 0.995i)T^{2}
67 1+(−0.445−0.895i)T2 1 + (-0.445 - 0.895i)T^{2}
71 1+(0.739−0.673i)T2 1 + (0.739 - 0.673i)T^{2}
73 1+(−0.576+0.435i)T+(0.273−0.961i)T2 1 + (-0.576 + 0.435i)T + (0.273 - 0.961i)T^{2}
79 1+(0.445+0.895i)T2 1 + (0.445 + 0.895i)T^{2}
83 1+(−0.932−0.361i)T2 1 + (-0.932 - 0.361i)T^{2}
89 1+(−0.156+1.69i)T+(−0.982−0.183i)T2 1 + (-0.156 + 1.69i)T + (-0.982 - 0.183i)T^{2}
97 1+(0.646+1.66i)T+(−0.739+0.673i)T2 1 + (0.646 + 1.66i)T + (-0.739 + 0.673i)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

−9.856489586531812806575103345194, −8.683405276725889810948310710926, −7.88210896094076673316739811203, −7.32373339080767102461169709476, −6.05458712102985994661277470431, −5.08071819873882782045442033978, −4.41177985844225435506453738981, −3.31560958462054778070525265224, −2.77201516152396707637811786059, −0.54864881468592047112449689459, 2.46460298854711158918327676560, 3.77820371680408745594077474261, 4.02314947690434786689823932411, 5.06457843295597270247734119611, 6.33739812708424034119789978613, 6.97125656628177057379997644457, 7.70661562925226833259022172991, 8.569493632648885459607317843118, 9.045390343664980299117188190740, 10.86703626645144183142887297186

Graph of the ZZ-function along the critical line