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Properties

Label 2-3750-1.1-c1-0-53
Degree 22
Conductor 37503750
Sign −1-1
Analytic cond. 29.943929.9439
Root an. cond. 5.472105.47210
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 4.16·7-s − 8-s + 9-s − 4.27·11-s + 12-s + 6.14·13-s + 4.16·14-s + 16-s − 5.15·17-s − 18-s + 7.61·19-s − 4.16·21-s + 4.27·22-s + 2.99·23-s − 24-s − 6.14·26-s + 27-s − 4.16·28-s − 5.67·29-s + 3.45·31-s − 32-s − 4.27·33-s + 5.15·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.57·7-s − 0.353·8-s + 0.333·9-s − 1.28·11-s + 0.288·12-s + 1.70·13-s + 1.11·14-s + 0.250·16-s − 1.25·17-s − 0.235·18-s + 1.74·19-s − 0.908·21-s + 0.910·22-s + 0.623·23-s − 0.204·24-s − 1.20·26-s + 0.192·27-s − 0.787·28-s − 1.05·29-s + 0.620·31-s − 0.176·32-s − 0.743·33-s + 0.884·34-s + ⋯

Functional equation

Λ(s)=(3750s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3750s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37503750    =    2⋅3⋅542 \cdot 3 \cdot 5^{4}
Sign: −1-1
Analytic conductor: 29.943929.9439
Root analytic conductor: 5.472105.47210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3750, ( :1/2), −1)(2,\ 3750,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+T 1 + T
3 1−T 1 - T
5 1 1
good7 1+4.16T+7T2 1 + 4.16T + 7T^{2}
11 1+4.27T+11T2 1 + 4.27T + 11T^{2}
13 1−6.14T+13T2 1 - 6.14T + 13T^{2}
17 1+5.15T+17T2 1 + 5.15T + 17T^{2}
19 1−7.61T+19T2 1 - 7.61T + 19T^{2}
23 1−2.99T+23T2 1 - 2.99T + 23T^{2}
29 1+5.67T+29T2 1 + 5.67T + 29T^{2}
31 1−3.45T+31T2 1 - 3.45T + 31T^{2}
37 1+7.53T+37T2 1 + 7.53T + 37T^{2}
41 1−11.0T+41T2 1 - 11.0T + 41T^{2}
43 1+3.38T+43T2 1 + 3.38T + 43T^{2}
47 1−0.632T+47T2 1 - 0.632T + 47T^{2}
53 1+6.34T+53T2 1 + 6.34T + 53T^{2}
59 1+3.03T+59T2 1 + 3.03T + 59T^{2}
61 1+2.29T+61T2 1 + 2.29T + 61T^{2}
67 1+0.726T+67T2 1 + 0.726T + 67T^{2}
71 1−3.20T+71T2 1 - 3.20T + 71T^{2}
73 1+9.06T+73T2 1 + 9.06T + 73T^{2}
79 1+6.62T+79T2 1 + 6.62T + 79T^{2}
83 1+3.20T+83T2 1 + 3.20T + 83T^{2}
89 1+4.59T+89T2 1 + 4.59T + 89T^{2}
97 1+9.16T+97T2 1 + 9.16T + 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

−8.244324049671434519196746979335, −7.47185942179522490520767387354, −6.82499043928895885799975758099, −6.08760810541827916461393470862, −5.33238745755257280809383748619, −3.99130612133991251850097840226, −3.16403325876517513982872954565, −2.68001203199375485629480820539, −1.33218282045158082819987072289, 0, 1.33218282045158082819987072289, 2.68001203199375485629480820539, 3.16403325876517513982872954565, 3.99130612133991251850097840226, 5.33238745755257280809383748619, 6.08760810541827916461393470862, 6.82499043928895885799975758099, 7.47185942179522490520767387354, 8.244324049671434519196746979335

Graph of the ZZ-function along the critical line