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Properties

Label 2-700-28.27-c1-0-13
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  + (−0.366 − 1.36i)2-s − 1.73·3-s + (−1.73 + i)4-s + (0.633 + 2.36i)6-s + (−1.73 − 2i)7-s + (2 + 1.99i)8-s + 0.267i·11-s + (2.99 − 1.73i)12-s + 0.464i·13-s + (−2.09 + 3.09i)14-s + (1.99 − 3.46i)16-s + 6.46i·17-s − 6·19-s + (2.99 + 3.46i)21-s + (0.366 − 0.0980i)22-s − 1.46i·23-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s − 1.00·3-s + (−0.866 + 0.5i)4-s + (0.258 + 0.965i)6-s + (−0.654 − 0.755i)7-s + (0.707 + 0.707i)8-s + 0.0807i·11-s + (0.866 − 0.499i)12-s + 0.128i·13-s + (−0.560 + 0.827i)14-s + (0.499 − 0.866i)16-s + 1.56i·17-s − 1.37·19-s + (0.654 + 0.755i)21-s + (0.0780 − 0.0209i)22-s − 0.305i·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

\(L(1)\) \(\approx\) \(0.566473 - 0.0953369i\)
\(L(\frac12)\) \(\approx\) \(0.566473 - 0.0953369i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
5 \( 1 \)
7 \( 1 + (1.73 + 2i)T \)
good3 \( 1 + 1.73T + 3T^{2} \)
11 \( 1 - 0.267iT - 11T^{2} \)
13 \( 1 - 0.464iT - 13T^{2} \)
17 \( 1 - 6.46iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 - 7.92T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 9.46T + 37T^{2} \)
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 \( 1 - 9.46iT - 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 - 7.46iT - 71T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 - 14.6iT - 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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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

−10.50902797929184277072667839479, −9.998059493193729099519372654609, −8.764687198701634143337064636817, −8.053008855433997247202009104733, −6.67450551672930849786619109366, −6.00145135498016421227176469775, −4.63197027447726963819828443750, −3.93191817566789420835583005662, −2.55836021125705032684102506303, −0.899508843648121097689387714738, 0.53070849022740900579759320909, 2.80762299733371432146553395012, 4.53590527265417600062382932324, 5.26038683804315955313386208851, 6.32267782374629733809024737040, 6.51939265017407653763431927850, 7.86023260382316220329521041059, 8.734233910293262493093276743107, 9.550920362884854476395566086113, 10.33575888662204612586371631720

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