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Properties

Label 2-700-20.3-c1-0-46
Degree $2$
Conductor $700$
Sign $-0.909 + 0.414i$
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  + (−1.26 − 0.622i)2-s + (2.09 − 2.09i)3-s + (1.22 + 1.58i)4-s + (−3.96 + 1.35i)6-s + (−0.707 − 0.707i)7-s + (−0.572 − 2.76i)8-s − 5.78i·9-s + 0.214i·11-s + (5.88 + 0.744i)12-s + (−4.29 − 4.29i)13-s + (0.457 + 1.33i)14-s + (−0.996 + 3.87i)16-s + (−2.00 + 2.00i)17-s + (−3.60 + 7.34i)18-s + 0.877·19-s + ⋯
L(s)  = 1  + (−0.897 − 0.440i)2-s + (1.21 − 1.21i)3-s + (0.612 + 0.790i)4-s + (−1.61 + 0.554i)6-s + (−0.267 − 0.267i)7-s + (−0.202 − 0.979i)8-s − 1.92i·9-s + 0.0648i·11-s + (1.69 + 0.214i)12-s + (−1.19 − 1.19i)13-s + (0.122 + 0.357i)14-s + (−0.249 + 0.968i)16-s + (−0.487 + 0.487i)17-s + (−0.848 + 1.73i)18-s + 0.201·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.909 + 0.414i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.909 + 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255564 - 1.17696i\)
\(L(\frac12)\) \(\approx\) \(0.255564 - 1.17696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.622i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-2.09 + 2.09i)T - 3iT^{2} \)
11 \( 1 - 0.214iT - 11T^{2} \)
13 \( 1 + (4.29 + 4.29i)T + 13iT^{2} \)
17 \( 1 + (2.00 - 2.00i)T - 17iT^{2} \)
19 \( 1 - 0.877T + 19T^{2} \)
23 \( 1 + (-0.0902 + 0.0902i)T - 23iT^{2} \)
29 \( 1 + 4.03iT - 29T^{2} \)
31 \( 1 + 8.60iT - 31T^{2} \)
37 \( 1 + (-1.29 + 1.29i)T - 37iT^{2} \)
41 \( 1 - 2.91T + 41T^{2} \)
43 \( 1 + (-2.06 + 2.06i)T - 43iT^{2} \)
47 \( 1 + (-4.88 - 4.88i)T + 47iT^{2} \)
53 \( 1 + (-2.77 - 2.77i)T + 53iT^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 8.32T + 61T^{2} \)
67 \( 1 + (-0.555 - 0.555i)T + 67iT^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 + (4.18 + 4.18i)T + 73iT^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + (5.30 - 5.30i)T - 83iT^{2} \)
89 \( 1 - 12.7iT - 89T^{2} \)
97 \( 1 + (-2.58 + 2.58i)T - 97iT^{2} \)
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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

−9.815891382474382556422133680725, −9.204010175252619078999147497188, −8.190914814839708322714758953247, −7.68403422779183602494401148262, −7.07356982471399074260782709676, −6.00372448025121420542136743288, −4.01147520634835495286125863195, −2.83227310218340898564831021446, −2.19509873324764539628995496702, −0.69923683080143039503293027531, 2.11067073280525694492025498609, 3.05535563312359669559046172503, 4.45817599945576207548763654134, 5.27458946932277514251095444625, 6.75543496834180853521429897973, 7.50129482971574124858322607008, 8.646839973729013837079557213906, 9.047983764501068830304810465241, 9.728314222235385953293676939690, 10.35108728628896726870201287191

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