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Properties

Label 2-750-1.1-c1-0-7
Degree 22
Conductor 750750
Sign 11
Analytic cond. 5.988785.98878
Root an. cond. 2.447192.44719
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 4.61·7-s − 8-s + 9-s + 0.854·11-s + 12-s + 5.61·13-s − 4.61·14-s + 16-s − 5.70·17-s − 18-s − 7.09·19-s + 4.61·21-s − 0.854·22-s + 8.09·23-s − 24-s − 5.61·26-s + 27-s + 4.61·28-s − 7.70·29-s + 2.47·31-s − 32-s + 0.854·33-s + 5.70·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.74·7-s − 0.353·8-s + 0.333·9-s + 0.257·11-s + 0.288·12-s + 1.55·13-s − 1.23·14-s + 0.250·16-s − 1.38·17-s − 0.235·18-s − 1.62·19-s + 1.00·21-s − 0.182·22-s + 1.68·23-s − 0.204·24-s − 1.10·26-s + 0.192·27-s + 0.872·28-s − 1.43·29-s + 0.444·31-s − 0.176·32-s + 0.148·33-s + 0.978·34-s + ⋯

Functional equation

Λ(s)=(750s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(750s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 750750    =    2⋅3⋅532 \cdot 3 \cdot 5^{3}
Sign: 11
Analytic conductor: 5.988785.98878
Root analytic conductor: 2.447192.44719
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 750, ( :1/2), 1)(2,\ 750,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6856991161.685699116
L(12)L(\frac12) ≈\approx 1.6856991161.685699116
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 1−4.61T+7T2 1 - 4.61T + 7T^{2}
11 1−0.854T+11T2 1 - 0.854T + 11T^{2}
13 1−5.61T+13T2 1 - 5.61T + 13T^{2}
17 1+5.70T+17T2 1 + 5.70T + 17T^{2}
19 1+7.09T+19T2 1 + 7.09T + 19T^{2}
23 1−8.09T+23T2 1 - 8.09T + 23T^{2}
29 1+7.70T+29T2 1 + 7.70T + 29T^{2}
31 1−2.47T+31T2 1 - 2.47T + 31T^{2}
37 1−0.618T+37T2 1 - 0.618T + 37T^{2}
41 1−2.61T+41T2 1 - 2.61T + 41T^{2}
43 1+5.70T+43T2 1 + 5.70T + 43T^{2}
47 1−6.38T+47T2 1 - 6.38T + 47T^{2}
53 1−2.61T+53T2 1 - 2.61T + 53T^{2}
59 1+4.14T+59T2 1 + 4.14T + 59T^{2}
61 1+1.70T+61T2 1 + 1.70T + 61T^{2}
67 1+1.23T+67T2 1 + 1.23T + 67T^{2}
71 1−4.47T+71T2 1 - 4.47T + 71T^{2}
73 1+8T+73T2 1 + 8T + 73T^{2}
79 1−3.52T+79T2 1 - 3.52T + 79T^{2}
83 1−2T+83T2 1 - 2T + 83T^{2}
89 1+4.56T+89T2 1 + 4.56T + 89T^{2}
97 1−8.18T+97T2 1 - 8.18T + 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

−10.60117460001814790123815808961, −8.972527151304895505297455197288, −8.808798108697884325141960292324, −8.043164029822116798756954868928, −7.09696044061933819000435744988, −6.13507820152330042751533510841, −4.79444024909693425982103301951, −3.87869696057505651806647739525, −2.29213383794753257312806161143, −1.36708512983003916590438478009, 1.36708512983003916590438478009, 2.29213383794753257312806161143, 3.87869696057505651806647739525, 4.79444024909693425982103301951, 6.13507820152330042751533510841, 7.09696044061933819000435744988, 8.043164029822116798756954868928, 8.808798108697884325141960292324, 8.972527151304895505297455197288, 10.60117460001814790123815808961

Graph of the ZZ-function along the critical line