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Properties

Label 2-156-13.10-c3-0-4
Degree 22
Conductor 156156
Sign −0.923+0.383i-0.923 + 0.383i
Analytic cond. 9.204299.20429
Root an. cond. 3.033853.03385
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 2.59i)3-s − 11.8i·5-s + (−19.2 − 11.1i)7-s + (−4.5 + 7.79i)9-s + (−49.0 + 28.3i)11-s + (−28.0 + 37.5i)13-s + (30.9 − 17.8i)15-s + (20.5 − 35.6i)17-s + (−99.4 − 57.4i)19-s − 66.6i·21-s + (−33.8 − 58.6i)23-s − 16.5·25-s − 27·27-s + (67.1 + 116. i)29-s − 265. i·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 1.06i·5-s + (−1.03 − 0.600i)7-s + (−0.166 + 0.288i)9-s + (−1.34 + 0.776i)11-s + (−0.598 + 0.801i)13-s + (0.532 − 0.307i)15-s + (0.293 − 0.508i)17-s + (−1.20 − 0.693i)19-s − 0.692i·21-s + (−0.306 − 0.531i)23-s − 0.132·25-s − 0.192·27-s + (0.429 + 0.744i)29-s − 1.54i·31-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=((−0.923+0.383i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s+3/2) L(s)=((−0.923+0.383i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: −0.923+0.383i-0.923 + 0.383i
Analytic conductor: 9.204299.20429
Root analytic conductor: 3.033853.03385
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ156(49,⋅)\chi_{156} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 156, ( :3/2), −0.923+0.383i)(2,\ 156,\ (\ :3/2),\ -0.923 + 0.383i)

Particular Values

L(2)L(2) ≈\approx 0.0681430−0.342227i0.0681430 - 0.342227i
L(12)L(\frac12) ≈\approx 0.0681430−0.342227i0.0681430 - 0.342227i
L(52)L(\frac{5}{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)
bad2 1 1
3 1+(−1.5−2.59i)T 1 + (-1.5 - 2.59i)T
13 1+(28.0−37.5i)T 1 + (28.0 - 37.5i)T
good5 1+11.8iT−125T2 1 + 11.8iT - 125T^{2}
7 1+(19.2+11.1i)T+(171.5+297.i)T2 1 + (19.2 + 11.1i)T + (171.5 + 297. i)T^{2}
11 1+(49.0−28.3i)T+(665.5−1.15e3i)T2 1 + (49.0 - 28.3i)T + (665.5 - 1.15e3i)T^{2}
17 1+(−20.5+35.6i)T+(−2.45e3−4.25e3i)T2 1 + (-20.5 + 35.6i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(99.4+57.4i)T+(3.42e3+5.94e3i)T2 1 + (99.4 + 57.4i)T + (3.42e3 + 5.94e3i)T^{2}
23 1+(33.8+58.6i)T+(−6.08e3+1.05e4i)T2 1 + (33.8 + 58.6i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(−67.1−116.i)T+(−1.21e4+2.11e4i)T2 1 + (-67.1 - 116. i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+265.iT−2.97e4T2 1 + 265. iT - 2.97e4T^{2}
37 1+(60.0−34.6i)T+(2.53e4−4.38e4i)T2 1 + (60.0 - 34.6i)T + (2.53e4 - 4.38e4i)T^{2}
41 1+(99.0−57.2i)T+(3.44e4−5.96e4i)T2 1 + (99.0 - 57.2i)T + (3.44e4 - 5.96e4i)T^{2}
43 1+(−153.+266.i)T+(−3.97e4−6.88e4i)T2 1 + (-153. + 266. i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1−518.iT−1.03e5T2 1 - 518. iT - 1.03e5T^{2}
53 1−304.T+1.48e5T2 1 - 304.T + 1.48e5T^{2}
59 1+(608.+351.i)T+(1.02e5+1.77e5i)T2 1 + (608. + 351. i)T + (1.02e5 + 1.77e5i)T^{2}
61 1+(−324.+562.i)T+(−1.13e5−1.96e5i)T2 1 + (-324. + 562. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(622.−359.i)T+(1.50e5−2.60e5i)T2 1 + (622. - 359. i)T + (1.50e5 - 2.60e5i)T^{2}
71 1+(−782.−451.i)T+(1.78e5+3.09e5i)T2 1 + (-782. - 451. i)T + (1.78e5 + 3.09e5i)T^{2}
73 1+562.iT−3.89e5T2 1 + 562. iT - 3.89e5T^{2}
79 1−530.T+4.93e5T2 1 - 530.T + 4.93e5T^{2}
83 1−986.iT−5.71e5T2 1 - 986. iT - 5.71e5T^{2}
89 1+(271.−156.i)T+(3.52e5−6.10e5i)T2 1 + (271. - 156. i)T + (3.52e5 - 6.10e5i)T^{2}
97 1+(120.+69.5i)T+(4.56e5+7.90e5i)T2 1 + (120. + 69.5i)T + (4.56e5 + 7.90e5i)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

−12.36545759549883287782270115675, −10.79583597529669144567126954787, −9.854060851407813824004675382193, −9.139441971227137319028454329180, −7.929024435895338178212357791152, −6.73352810428569614319637625662, −5.07508647671346409852654154993, −4.23762912723302830297104910332, −2.50438578849629805766591237598, −0.14625104279669167634750769432, 2.52800787403769465469585627819, 3.32701636147664394754716206633, 5.61152269980539892657256239224, 6.49068326972838892968519756046, 7.67533212232075043105879321638, 8.623443492291399811647328070140, 10.14288655224837100589656174505, 10.63961858868116878290135619134, 12.15540750039775824348842829950, 12.88640963354841638479906791062

Graph of the ZZ-function along the critical line