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Properties

Label 2-700-20.3-c1-0-48
Degree $2$
Conductor $700$
Sign $0.281 + 0.959i$
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  + (1.36 − 0.361i)2-s + (1.26 − 1.26i)3-s + (1.73 − 0.988i)4-s + (1.27 − 2.18i)6-s + (−0.707 − 0.707i)7-s + (2.02 − 1.97i)8-s − 0.204i·9-s + 2.11i·11-s + (0.950 − 3.45i)12-s + (−2.86 − 2.86i)13-s + (−1.22 − 0.711i)14-s + (2.04 − 3.43i)16-s + (4.47 − 4.47i)17-s + (−0.0740 − 0.280i)18-s + 2.95·19-s + ⋯
L(s)  = 1  + (0.966 − 0.255i)2-s + (0.730 − 0.730i)3-s + (0.869 − 0.494i)4-s + (0.519 − 0.893i)6-s + (−0.267 − 0.267i)7-s + (0.714 − 0.699i)8-s − 0.0682i·9-s + 0.637i·11-s + (0.274 − 0.996i)12-s + (−0.794 − 0.794i)13-s + (−0.326 − 0.190i)14-s + (0.511 − 0.859i)16-s + (1.08 − 1.08i)17-s + (−0.0174 − 0.0660i)18-s + 0.677·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.281 + 0.959i$
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: \((2,\ 700,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68494 - 2.01145i\)
\(L(\frac12)\) \(\approx\) \(2.68494 - 2.01145i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.361i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-1.26 + 1.26i)T - 3iT^{2} \)
11 \( 1 - 2.11iT - 11T^{2} \)
13 \( 1 + (2.86 + 2.86i)T + 13iT^{2} \)
17 \( 1 + (-4.47 + 4.47i)T - 17iT^{2} \)
19 \( 1 - 2.95T + 19T^{2} \)
23 \( 1 + (3.94 - 3.94i)T - 23iT^{2} \)
29 \( 1 + 2.86iT - 29T^{2} \)
31 \( 1 - 6.01iT - 31T^{2} \)
37 \( 1 + (3.91 - 3.91i)T - 37iT^{2} \)
41 \( 1 + 7.98T + 41T^{2} \)
43 \( 1 + (5.64 - 5.64i)T - 43iT^{2} \)
47 \( 1 + (-4.30 - 4.30i)T + 47iT^{2} \)
53 \( 1 + (-0.561 - 0.561i)T + 53iT^{2} \)
59 \( 1 - 4.97T + 59T^{2} \)
61 \( 1 - 1.68T + 61T^{2} \)
67 \( 1 + (-8.06 - 8.06i)T + 67iT^{2} \)
71 \( 1 + 0.610iT - 71T^{2} \)
73 \( 1 + (-3.23 - 3.23i)T + 73iT^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + (5.51 - 5.51i)T - 83iT^{2} \)
89 \( 1 + 2.07iT - 89T^{2} \)
97 \( 1 + (-1.95 + 1.95i)T - 97iT^{2} \)
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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

−10.01885315564016236398830109585, −9.877625918760919174776532370565, −8.241897278029620847784655073497, −7.37777225190129368857698766180, −6.99685202888914820143768968252, −5.56864941021598566203484209052, −4.84148966232142673383260979193, −3.39526371863726875963183650119, −2.68169705482961991415917775253, −1.41661323105861444998198100336, 2.17611822741704575098929678753, 3.38165868725220388570033189531, 3.93547512547416849090670850574, 5.10805300863236571899125499471, 6.02847169242369158907176573546, 6.97747135964045610315077803651, 8.083001401498175903399831037569, 8.791116045044920932410197093402, 9.865401600955517785286933373813, 10.49843370501370759350674379425

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