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Properties

Label 2-700-28.27-c1-0-20
Degree $2$
Conductor $700$
Sign $-0.00212 - 0.999i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 + 0.331i)2-s + 2.13·3-s + (1.78 − 0.910i)4-s + (−2.93 + 0.707i)6-s + (−1.19 + 2.35i)7-s + (−2.14 + 1.84i)8-s + 1.56·9-s + 2.33i·11-s + (3.80 − 1.94i)12-s + 1.09i·13-s + (0.868 − 3.63i)14-s + (2.34 − 3.24i)16-s + 4.98i·17-s + (−2.14 + 0.516i)18-s − 2.57·19-s + ⋯
L(s)  = 1  + (−0.972 + 0.234i)2-s + 1.23·3-s + (0.890 − 0.455i)4-s + (−1.19 + 0.288i)6-s + (−0.453 + 0.891i)7-s + (−0.759 + 0.650i)8-s + 0.520·9-s + 0.703i·11-s + (1.09 − 0.561i)12-s + 0.302i·13-s + (0.232 − 0.972i)14-s + (0.585 − 0.810i)16-s + 1.20i·17-s + (−0.506 + 0.121i)18-s − 0.590·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00212 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00212 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.00212 - 0.999i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.00212 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.875899 + 0.877759i\)
\(L(\frac12)\) \(\approx\) \(0.875899 + 0.877759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.331i)T \)
5 \( 1 \)
7 \( 1 + (1.19 - 2.35i)T \)
good3 \( 1 - 2.13T + 3T^{2} \)
11 \( 1 - 2.33iT - 11T^{2} \)
13 \( 1 - 1.09iT - 13T^{2} \)
17 \( 1 - 4.98iT - 17T^{2} \)
19 \( 1 + 2.57T + 19T^{2} \)
23 \( 1 - 6.04iT - 23T^{2} \)
29 \( 1 + 0.561T + 29T^{2} \)
31 \( 1 - 6.59T + 31T^{2} \)
37 \( 1 - 5.49T + 37T^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + 1.32iT - 43T^{2} \)
47 \( 1 - 9.74T + 47T^{2} \)
53 \( 1 + 8.58T + 53T^{2} \)
59 \( 1 - 14.3T + 59T^{2} \)
61 \( 1 + 0.620iT - 61T^{2} \)
67 \( 1 + 4.71iT - 67T^{2} \)
71 \( 1 - 11.9iT - 71T^{2} \)
73 \( 1 + 9.96iT - 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + 3.86T + 83T^{2} \)
89 \( 1 + 2.82iT - 89T^{2} \)
97 \( 1 + 14.9iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.24729143391807600868479962027, −9.559115557685312735248982324701, −8.895524310128927990911052855180, −8.286193582146877406479373803390, −7.47942331712600548571074924110, −6.46112629120117534181879593018, −5.54936535567884759848487624427, −3.87313193069943537686890387894, −2.64815646154556387458102835636, −1.85733291018775265205966047553, 0.75090791152708461574362963529, 2.53046452164211184632952479954, 3.16540342906410680854893423515, 4.31651790868033019658581153594, 6.15539754905489637565542219065, 7.04464382505090902528814819625, 7.938617337433091926617516323823, 8.491688350931095219795448390010, 9.358013610146822993277967555303, 10.00835239829657668090287748569

Graph of the $Z$-function along the critical line