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Properties

Label 2-700-28.27-c1-0-47
Degree $2$
Conductor $700$
Sign $0.377 - 0.925i$
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 + i)2-s + 2.44·3-s + 2i·4-s + (2.44 + 2.44i)6-s + (2.44 + i)7-s + (−2 + 2i)8-s + 2.99·9-s − 5i·11-s + 4.89i·12-s − 2.44i·13-s + (1.44 + 3.44i)14-s − 4·16-s + 4.89i·17-s + (2.99 + 2.99i)18-s + (5.99 + 2.44i)21-s + (5 − 5i)22-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.41·3-s + i·4-s + (0.999 + 0.999i)6-s + (0.925 + 0.377i)7-s + (−0.707 + 0.707i)8-s + 0.999·9-s − 1.50i·11-s + 1.41i·12-s − 0.679i·13-s + (0.387 + 0.921i)14-s − 16-s + 1.18i·17-s + (0.707 + 0.707i)18-s + (1.30 + 0.534i)21-s + (1.06 − 1.06i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.377 - 0.925i$
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: \((2,\ 700,\ (\ :1/2),\ 0.377 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.91561 + 1.95892i\)
\(L(\frac12)\) \(\approx\) \(2.91561 + 1.95892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 \)
7 \( 1 + (-2.44 - i)T \)
good3 \( 1 - 2.44T + 3T^{2} \)
11 \( 1 + 5iT - 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + iT - 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + 7.34T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 - 12.2iT - 41T^{2} \)
43 \( 1 + 11iT - 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 - 5iT - 71T^{2} \)
73 \( 1 + 2.44iT - 73T^{2} \)
79 \( 1 - 9iT - 79T^{2} \)
83 \( 1 - 2.44T + 83T^{2} \)
89 \( 1 + 2.44iT - 89T^{2} \)
97 \( 1 + 7.34iT - 97T^{2} \)
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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

−10.75593551605462487547874885358, −9.285050632753060567213796375886, −8.347015807579896558130561813104, −8.319982305709156021451232611984, −7.33603934988985543194578476251, −6.00823123341572054510092192746, −5.28466109350992982297749775141, −3.89539643778979726450819971188, −3.25462283823779950129963305013, −2.07287317658215946414262142072, 1.71604336752395738307051425510, 2.40132893601016522308064971155, 3.71281574786044364648898328731, 4.45691612973383970469121465518, 5.38005185479948006116091359946, 7.10453254970657288572452762871, 7.50050964154649040582427070656, 8.891716472945455591438007577648, 9.399363010199603488720948498370, 10.22715553737564465226667977626

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