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Properties

Label 2-3276-91.34-c1-0-18
Degree 22
Conductor 32763276
Sign 0.969+0.245i0.969 + 0.245i
Analytic cond. 26.158926.1589
Root an. cond. 5.114585.11458
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  + (−2.45 + 2.45i)5-s + (−1.75 − 1.97i)7-s + (−4.08 + 4.08i)11-s + (−2.80 − 2.27i)13-s − 1.70·17-s + (−3.16 + 3.16i)19-s − 8.05i·23-s − 7.05i·25-s − 0.784·29-s + (0.750 − 0.750i)31-s + (9.16 + 0.529i)35-s + (4.39 − 4.39i)37-s + (−0.897 + 0.897i)41-s + 7.57i·43-s + (−2.43 − 2.43i)47-s + ⋯
L(s)  = 1  + (−1.09 + 1.09i)5-s + (−0.665 − 0.746i)7-s + (−1.23 + 1.23i)11-s + (−0.776 − 0.629i)13-s − 0.413·17-s + (−0.726 + 0.726i)19-s − 1.67i·23-s − 1.41i·25-s − 0.145·29-s + (0.134 − 0.134i)31-s + (1.54 + 0.0895i)35-s + (0.723 − 0.723i)37-s + (−0.140 + 0.140i)41-s + 1.15i·43-s + (−0.354 − 0.354i)47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((0.969+0.245i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((0.969+0.245i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.969+0.245i0.969 + 0.245i
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3276(1945,⋅)\chi_{3276} (1945, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), 0.969+0.245i)(2,\ 3276,\ (\ :1/2),\ 0.969 + 0.245i)

Particular Values

L(1)L(1) ≈\approx 0.55609681040.5560968104
L(12)L(\frac12) ≈\approx 0.55609681040.5560968104
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)
bad2 1 1
3 1 1
7 1+(1.75+1.97i)T 1 + (1.75 + 1.97i)T
13 1+(2.80+2.27i)T 1 + (2.80 + 2.27i)T
good5 1+(2.45−2.45i)T−5iT2 1 + (2.45 - 2.45i)T - 5iT^{2}
11 1+(4.08−4.08i)T−11iT2 1 + (4.08 - 4.08i)T - 11iT^{2}
17 1+1.70T+17T2 1 + 1.70T + 17T^{2}
19 1+(3.16−3.16i)T−19iT2 1 + (3.16 - 3.16i)T - 19iT^{2}
23 1+8.05iT−23T2 1 + 8.05iT - 23T^{2}
29 1+0.784T+29T2 1 + 0.784T + 29T^{2}
31 1+(−0.750+0.750i)T−31iT2 1 + (-0.750 + 0.750i)T - 31iT^{2}
37 1+(−4.39+4.39i)T−37iT2 1 + (-4.39 + 4.39i)T - 37iT^{2}
41 1+(0.897−0.897i)T−41iT2 1 + (0.897 - 0.897i)T - 41iT^{2}
43 1−7.57iT−43T2 1 - 7.57iT - 43T^{2}
47 1+(2.43+2.43i)T+47iT2 1 + (2.43 + 2.43i)T + 47iT^{2}
53 1−6.24T+53T2 1 - 6.24T + 53T^{2}
59 1+(−5.29−5.29i)T+59iT2 1 + (-5.29 - 5.29i)T + 59iT^{2}
61 1−0.691iT−61T2 1 - 0.691iT - 61T^{2}
67 1+(−7.16−7.16i)T+67iT2 1 + (-7.16 - 7.16i)T + 67iT^{2}
71 1+(−11.1−11.1i)T+71iT2 1 + (-11.1 - 11.1i)T + 71iT^{2}
73 1+(−1.09−1.09i)T+73iT2 1 + (-1.09 - 1.09i)T + 73iT^{2}
79 1+7.14T+79T2 1 + 7.14T + 79T^{2}
83 1+(−3.09+3.09i)T−83iT2 1 + (-3.09 + 3.09i)T - 83iT^{2}
89 1+(8.20+8.20i)T+89iT2 1 + (8.20 + 8.20i)T + 89iT^{2}
97 1+(9.21−9.21i)T−97iT2 1 + (9.21 - 9.21i)T - 97iT^{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

−8.204069666798083456206710373852, −7.88748937197649150340548799967, −6.94662782961085287246419032946, −6.84046644997339271508295112399, −5.61083673598891722464010428844, −4.45087102278136783075812571847, −4.05016767119984673009072210209, −2.87596279491498457344742109845, −2.39657804375735325806557621442, −0.32320301992264541130044798433, 0.53939929680852307308973977036, 2.17372520600892221895660724563, 3.15306673317966416801561432801, 3.97282003470413442351668701769, 4.98972757988454184463659375620, 5.40094246788367066214255410725, 6.41344133581419601190880523928, 7.31819071829264252437879260341, 8.110070600920168448050795378790, 8.593157777475803677877522737933

Graph of the ZZ-function along the critical line