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Properties

Label 2-700-140.27-c0-0-1
Degree 22
Conductor 700700
Sign −0.229−0.973i-0.229 - 0.973i
Analytic cond. 0.3493450.349345
Root an. cond. 0.5910540.591054
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)2-s − 1.00i·4-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + i·9-s − 1.00i·14-s − 1.00·16-s + (−0.707 − 0.707i)18-s + (1.41 + 1.41i)23-s + (0.707 + 0.707i)28-s + 2i·29-s + (0.707 − 0.707i)32-s + 1.00·36-s + (−1.41 − 1.41i)43-s − 2.00·46-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + i·9-s − 1.00i·14-s − 1.00·16-s + (−0.707 − 0.707i)18-s + (1.41 + 1.41i)23-s + (0.707 + 0.707i)28-s + 2i·29-s + (0.707 − 0.707i)32-s + 1.00·36-s + (−1.41 − 1.41i)43-s − 2.00·46-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.229−0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s) L(s)=((−0.229−0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.229−0.973i-0.229 - 0.973i
Analytic conductor: 0.3493450.349345
Root analytic conductor: 0.5910540.591054
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ700(307,⋅)\chi_{700} (307, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :0), −0.229−0.973i)(2,\ 700,\ (\ :0),\ -0.229 - 0.973i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.58678677600.5867867760
L(12)L(\frac12) ≈\approx 0.58678677600.5867867760
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)
bad2 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
5 1 1
7 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good3 1−iT2 1 - iT^{2}
11 1−T2 1 - T^{2}
13 1+iT2 1 + iT^{2}
17 1−iT2 1 - iT^{2}
19 1−T2 1 - T^{2}
23 1+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)T + iT^{2}
29 1−2iT−T2 1 - 2iT - T^{2}
31 1+T2 1 + T^{2}
37 1+iT2 1 + iT^{2}
41 1−T2 1 - T^{2}
43 1+(1.41+1.41i)T+iT2 1 + (1.41 + 1.41i)T + iT^{2}
47 1+iT2 1 + iT^{2}
53 1−iT2 1 - iT^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1+(−1.41+1.41i)T−iT2 1 + (-1.41 + 1.41i)T - iT^{2}
71 1−T2 1 - T^{2}
73 1+iT2 1 + iT^{2}
79 1+T2 1 + T^{2}
83 1−iT2 1 - iT^{2}
89 1+T2 1 + T^{2}
97 1−iT2 1 - 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

−10.73098792285850473462189574181, −9.873750436357749173367187532386, −9.062762101882822741284211128927, −8.431442871835027251479942827423, −7.35232489365937480203590558845, −6.72670947749367994039517362325, −5.50586403046430644841154787085, −5.02992254519506161452014467640, −3.22320742186180687134708825688, −1.79710515612962831056229060091, 0.846763681165843152092747064034, 2.66049536488616773387402090552, 3.62684846329334024325044322582, 4.55594170247509726027221479847, 6.32901388526955899998722511304, 6.92689804270752575492875530616, 7.986476815199382154730814820442, 8.896642961291530397978538190496, 9.692394288438552140145991874321, 10.23809181413618453542343597327

Graph of the ZZ-function along the critical line