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Properties

Label 2-700-20.7-c1-0-48
Degree $2$
Conductor $700$
Sign $0.610 + 0.792i$
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  + (1.38 − 0.297i)2-s + (0.137 + 0.137i)3-s + (1.82 − 0.823i)4-s + (0.231 + 0.149i)6-s + (0.707 − 0.707i)7-s + (2.27 − 1.68i)8-s − 2.96i·9-s − 3.12i·11-s + (0.364 + 0.137i)12-s + (−2.50 + 2.50i)13-s + (0.766 − 1.18i)14-s + (2.64 − 3.00i)16-s + (−2.85 − 2.85i)17-s + (−0.882 − 4.09i)18-s + 3.29·19-s + ⋯
L(s)  = 1  + (0.977 − 0.210i)2-s + (0.0796 + 0.0796i)3-s + (0.911 − 0.411i)4-s + (0.0945 + 0.0610i)6-s + (0.267 − 0.267i)7-s + (0.804 − 0.594i)8-s − 0.987i·9-s − 0.942i·11-s + (0.105 + 0.0397i)12-s + (−0.694 + 0.694i)13-s + (0.204 − 0.317i)14-s + (0.660 − 0.750i)16-s + (−0.693 − 0.693i)17-s + (−0.207 − 0.965i)18-s + 0.755·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.610 + 0.792i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.610 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59674 - 1.27783i\)
\(L(\frac12)\) \(\approx\) \(2.59674 - 1.27783i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.297i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.137 - 0.137i)T + 3iT^{2} \)
11 \( 1 + 3.12iT - 11T^{2} \)
13 \( 1 + (2.50 - 2.50i)T - 13iT^{2} \)
17 \( 1 + (2.85 + 2.85i)T + 17iT^{2} \)
19 \( 1 - 3.29T + 19T^{2} \)
23 \( 1 + (-6.43 - 6.43i)T + 23iT^{2} \)
29 \( 1 - 3.55iT - 29T^{2} \)
31 \( 1 - 8.67iT - 31T^{2} \)
37 \( 1 + (3.64 + 3.64i)T + 37iT^{2} \)
41 \( 1 - 1.66T + 41T^{2} \)
43 \( 1 + (1.17 + 1.17i)T + 43iT^{2} \)
47 \( 1 + (0.598 - 0.598i)T - 47iT^{2} \)
53 \( 1 + (-1.12 + 1.12i)T - 53iT^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 6.81T + 61T^{2} \)
67 \( 1 + (6.83 - 6.83i)T - 67iT^{2} \)
71 \( 1 - 3.15iT - 71T^{2} \)
73 \( 1 + (6.00 - 6.00i)T - 73iT^{2} \)
79 \( 1 - 0.153T + 79T^{2} \)
83 \( 1 + (8.57 + 8.57i)T + 83iT^{2} \)
89 \( 1 - 17.4iT - 89T^{2} \)
97 \( 1 + (2.28 + 2.28i)T + 97iT^{2} \)
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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

−10.55847393288524662156000169600, −9.488833672637762234247288242289, −8.785943574224296964877921742877, −7.22825779424951457817747538785, −6.87263175356537358856644551511, −5.60908770559719014525671841605, −4.84783870173147238548787283568, −3.68021551294183394270981492219, −2.90044947248585648993954384379, −1.24782617654244604267013455101, 2.01090579860661090189015036186, 2.86139173778983306059960771672, 4.42231530956721153246636409099, 4.96057176649795204322234178964, 5.96933698111440768693366556815, 7.08866003998861675992459726621, 7.72849804235881304019928281763, 8.591714557836549233760051012243, 9.944297583628399942010117772422, 10.72515531827809086931091362771

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