Properties

Label 2-700-28.27-c1-0-29
Degree $2$
Conductor $700$
Sign $-0.188 - 0.981i$
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  + (−0.366 + 1.36i)2-s + 1.73·3-s + (−1.73 − i)4-s + (−0.633 + 2.36i)6-s + (1.73 + 2i)7-s + (2 − 1.99i)8-s − 0.267i·11-s + (−2.99 − 1.73i)12-s + 0.464i·13-s + (−3.36 + 1.63i)14-s + (1.99 + 3.46i)16-s + 6.46i·17-s + 6·19-s + (2.99 + 3.46i)21-s + (0.366 + 0.0980i)22-s + 1.46i·23-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + 1.00·3-s + (−0.866 − 0.5i)4-s + (−0.258 + 0.965i)6-s + (0.654 + 0.755i)7-s + (0.707 − 0.707i)8-s − 0.0807i·11-s + (−0.866 − 0.499i)12-s + 0.128i·13-s + (−0.899 + 0.436i)14-s + (0.499 + 0.866i)16-s + 1.56i·17-s + 1.37·19-s + (0.654 + 0.755i)21-s + (0.0780 + 0.0209i)22-s + 0.305i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.188 - 0.981i$
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: \((2,\ 700,\ (\ :1/2),\ -0.188 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11575 + 1.35095i\)
\(L(\frac12)\) \(\approx\) \(1.11575 + 1.35095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
5 \( 1 \)
7 \( 1 + (-1.73 - 2i)T \)
good3 \( 1 - 1.73T + 3T^{2} \)
11 \( 1 + 0.267iT - 11T^{2} \)
13 \( 1 - 0.464iT - 13T^{2} \)
17 \( 1 - 6.46iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 1.46iT - 23T^{2} \)
29 \( 1 - 7.92T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 9.46T + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 1.73T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 9.46iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + 7.46iT - 71T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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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

−10.41078933293567975975092713160, −9.381997318406651968004706958065, −8.840180632416924744407757954255, −8.067789849262556164493590586349, −7.57044891783568918210988783292, −6.21501182523023839745274184099, −5.48761035990834622754531498287, −4.35005406136056641150195919797, −3.13490689453419752830703271468, −1.64720153014001527606567993930, 1.01518454152294140742474394079, 2.49704480360999234001562125660, 3.28058496071971325801247671847, 4.39834078297719984233271961274, 5.30854930963982759558531380329, 7.16543114526691949933733476125, 7.84903429908166203145550050108, 8.573748920913104722376014385567, 9.512250599056241966927980142268, 9.991004052922898728737488012200

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