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Properties

Label 2-700-28.27-c1-0-57
Degree $2$
Conductor $700$
Sign $-0.766 + 0.642i$
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  + (−1.28 + 0.599i)2-s + 0.936·3-s + (1.28 − 1.53i)4-s + (−1.19 + 0.561i)6-s + (−2.60 − 0.468i)7-s + (−0.719 + 2.73i)8-s − 2.12·9-s + 2.39i·11-s + (1.19 − 1.43i)12-s + 2i·13-s + (3.61 − 0.961i)14-s + (−0.719 − 3.93i)16-s − 7.12i·17-s + (2.71 − 1.27i)18-s − 2.39·19-s + ⋯
L(s)  = 1  + (−0.905 + 0.424i)2-s + 0.540·3-s + (0.640 − 0.768i)4-s + (−0.489 + 0.229i)6-s + (−0.984 − 0.176i)7-s + (−0.254 + 0.967i)8-s − 0.707·9-s + 0.723i·11-s + (0.346 − 0.415i)12-s + 0.554i·13-s + (0.966 − 0.257i)14-s + (−0.179 − 0.983i)16-s − 1.72i·17-s + (0.640 − 0.300i)18-s − 0.550·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.766 + 0.642i$
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: \((2,\ 700,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0591618 - 0.162607i\)
\(L(\frac12)\) \(\approx\) \(0.0591618 - 0.162607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.599i)T \)
5 \( 1 \)
7 \( 1 + (2.60 + 0.468i)T \)
good3 \( 1 - 0.936T + 3T^{2} \)
11 \( 1 - 2.39iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 7.12iT - 17T^{2} \)
19 \( 1 + 2.39T + 19T^{2} \)
23 \( 1 + 5.73iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 6.67T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 7.12iT - 41T^{2} \)
43 \( 1 - 7.60iT - 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 14.2iT - 67T^{2} \)
71 \( 1 - 6.14iT - 71T^{2} \)
73 \( 1 + 9.36iT - 73T^{2} \)
79 \( 1 + 4.27iT - 79T^{2} \)
83 \( 1 - 0.936T + 83T^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 - 7.12iT - 97T^{2} \)
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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

−9.747209405155809379730126490836, −9.264944451063708481536130401891, −8.563126138511666196589051318578, −7.44990405334888073478691130222, −6.85075498039820134836233146704, −5.91723441277130935343167585474, −4.70420654154096881740470102903, −3.13396436469088455506952695584, −2.14833821210648465745823921525, −0.10359120068309105927198720120, 1.87037708464943446650511300860, 3.20515334990118541949177823544, 3.64350695603946682049097295056, 5.72588590158644156363180495571, 6.42038844385023770905624300137, 7.66855105061724272855460996682, 8.383452578898979290145517156464, 9.037035496902504769525352515252, 9.803593368978489483545246707774, 10.71731673283708804212924949536

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