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Properties

Label 2-867-51.47-c0-0-1
Degree 22
Conductor 867867
Sign 0.788−0.615i0.788 - 0.615i
Analytic cond. 0.4326890.432689
Root an. cond. 0.6577910.657791
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  + (0.707 + 0.707i)3-s − 4-s + (0.707 − 0.707i)7-s + 1.00i·9-s + (−0.707 − 0.707i)12-s + 13-s + 16-s + i·19-s + 1.00·21-s + i·25-s + (−0.707 + 0.707i)27-s + (−0.707 + 0.707i)28-s + (−0.707 − 0.707i)31-s − 1.00i·36-s + (−0.707 − 0.707i)37-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s − 4-s + (0.707 − 0.707i)7-s + 1.00i·9-s + (−0.707 − 0.707i)12-s + 13-s + 16-s + i·19-s + 1.00·21-s + i·25-s + (−0.707 + 0.707i)27-s + (−0.707 + 0.707i)28-s + (−0.707 − 0.707i)31-s − 1.00i·36-s + (−0.707 − 0.707i)37-s + ⋯

Functional equation

Λ(s)=(867s/2 ΓC(s) L(s)=((0.788−0.615i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(867s/2 ΓC(s) L(s)=((0.788−0.615i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 867867    =    3⋅1723 \cdot 17^{2}
Sign: 0.788−0.615i0.788 - 0.615i
Analytic conductor: 0.4326890.432689
Root analytic conductor: 0.6577910.657791
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ867(251,⋅)\chi_{867} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 867, ( :0), 0.788−0.615i)(2,\ 867,\ (\ :0),\ 0.788 - 0.615i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0998416451.099841645
L(12)L(\frac12) ≈\approx 1.0998416451.099841645
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)
bad3 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
17 1 1
good2 1+T2 1 + T^{2}
5 1−iT2 1 - iT^{2}
7 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
11 1+iT2 1 + iT^{2}
13 1−T+T2 1 - T + T^{2}
19 1−iT−T2 1 - iT - T^{2}
23 1+iT2 1 + iT^{2}
29 1−iT2 1 - iT^{2}
31 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
37 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
41 1+iT2 1 + iT^{2}
43 1+iT−T2 1 + iT - T^{2}
47 1−T2 1 - T^{2}
53 1+T2 1 + T^{2}
59 1+T2 1 + T^{2}
61 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
67 1+T+T2 1 + T + T^{2}
71 1−iT2 1 - iT^{2}
73 1+(1.41+1.41i)T+iT2 1 + (1.41 + 1.41i)T + iT^{2}
79 1+(−1.41+1.41i)T−iT2 1 + (-1.41 + 1.41i)T - iT^{2}
83 1+T2 1 + T^{2}
89 1−T2 1 - T^{2}
97 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{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

−10.48508039029054972955687261241, −9.456885628190360000703557496621, −8.864628655111482119643538539627, −8.056377028364969974934687296803, −7.44260067109104452478358349948, −5.81377201354538384911422407052, −4.96110723840541831610336365924, −3.98667549045219346519891785970, −3.50458912913076591424491464664, −1.64694741879472401837446572265, 1.35190921737122512040792026496, 2.73495878606900186375558057308, 3.87004982682175354458229525103, 4.92373286025177504971815052936, 5.91396263915775707123898883969, 6.93877467487005068763566155171, 8.068490484758100845757276737252, 8.597649720194864544514793408364, 9.075784581496676114162104654934, 10.07816280356452638521178046872

Graph of the ZZ-function along the critical line