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Properties

Label 2-700-28.27-c1-0-59
Degree $2$
Conductor $700$
Sign $0.944 - 0.327i$
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 + 0.366i)2-s + 1.73·3-s + (1.73 + i)4-s + (2.36 + 0.633i)6-s + (1.73 − 2i)7-s + (1.99 + 2i)8-s + 3.73i·11-s + (2.99 + 1.73i)12-s − 6.46i·13-s + (3.09 − 2.09i)14-s + (1.99 + 3.46i)16-s − 0.464i·17-s − 6·19-s + (2.99 − 3.46i)21-s + (−1.36 + 5.09i)22-s + 5.46i·23-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + 1.00·3-s + (0.866 + 0.5i)4-s + (0.965 + 0.258i)6-s + (0.654 − 0.755i)7-s + (0.707 + 0.707i)8-s + 1.12i·11-s + (0.866 + 0.499i)12-s − 1.79i·13-s + (0.827 − 0.560i)14-s + (0.499 + 0.866i)16-s − 0.112i·17-s − 1.37·19-s + (0.654 − 0.755i)21-s + (−0.291 + 1.08i)22-s + 1.13i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.944 - 0.327i$
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: \((2,\ 700,\ (\ :1/2),\ 0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.70457 + 0.623477i\)
\(L(\frac12)\) \(\approx\) \(3.70457 + 0.623477i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
5 \( 1 \)
7 \( 1 + (-1.73 + 2i)T \)
good3 \( 1 - 1.73T + 3T^{2} \)
11 \( 1 - 3.73iT - 11T^{2} \)
13 \( 1 + 6.46iT - 13T^{2} \)
17 \( 1 + 0.464iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 5.46iT - 23T^{2} \)
29 \( 1 + 5.92T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 2.53T + 37T^{2} \)
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 \( 1 - 2.53iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 0.535iT - 71T^{2} \)
73 \( 1 - 0.928iT - 73T^{2} \)
79 \( 1 + 2.66iT - 79T^{2} \)
83 \( 1 - 8.53T + 83T^{2} \)
89 \( 1 - 9.46iT - 89T^{2} \)
97 \( 1 + 7.39iT - 97T^{2} \)
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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

−10.59492502093965585698316281218, −9.708035042480779230503978716567, −8.348712549126064641539871242972, −7.83957040714112867614551492601, −7.17139850154194940540557527771, −5.89901175335517531165153752718, −4.88968005894662091227957285750, −3.96125512411985533405767238561, −2.99429481514725272822024367170, −1.89038467419989147946381234588, 1.90697823035968377339251657801, 2.64100216716250976032637645858, 3.84819847360057437746789104349, 4.68232537050479925388101802253, 5.93118694838157245823471270744, 6.60897355234665664644116675271, 7.954691204988293220615382023241, 8.692332931386267724098766806575, 9.351406033392508636877240228739, 10.70422805559011278369826445666

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