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Properties

Label 2-700-7.4-c1-0-7
Degree $2$
Conductor $700$
Sign $0.991 - 0.126i$
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  + (−1.5 + 2.59i)3-s + (−0.5 − 2.59i)7-s + (−3 − 5.19i)9-s + (1 − 1.73i)11-s + 6·13-s + (1 − 1.73i)17-s + (7.5 + 2.59i)21-s + (−4.5 − 7.79i)23-s + 9·27-s + 3·29-s + (−1 + 1.73i)31-s + (3 + 5.19i)33-s + (4 + 6.92i)37-s + (−9 + 15.5i)39-s + 5·41-s + ⋯
L(s)  = 1  + (−0.866 + 1.49i)3-s + (−0.188 − 0.981i)7-s + (−1 − 1.73i)9-s + (0.301 − 0.522i)11-s + 1.66·13-s + (0.242 − 0.420i)17-s + (1.63 + 0.566i)21-s + (−0.938 − 1.62i)23-s + 1.73·27-s + 0.557·29-s + (−0.179 + 0.311i)31-s + (0.522 + 0.904i)33-s + (0.657 + 1.13i)37-s + (−1.44 + 2.49i)39-s + 0.780·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07337 + 0.0681167i\)
\(L(\frac12)\) \(\approx\) \(1.07337 + 0.0681167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.5 + 2.59i)T \)
good3 \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.54802795130702890349441450560, −9.862395671494668382720977579197, −8.945986645573651820304592625124, −8.063316455681579894919765870180, −6.48513338097050374821343449940, −6.06971771051922446566618561723, −4.80540911759818867226455012828, −4.06500472574838242289754385963, −3.29358468842960578616381354387, −0.75743476891547149038775290396, 1.23548129594604909976814531111, 2.26010940022169707386274187625, 3.86366605373192957546620990107, 5.60967500654854610223777264096, 5.88680034764386969265989558344, 6.79159175558741715181563945776, 7.72564588396874043121645627425, 8.501938853220002993874118682912, 9.486373957961796894308117388180, 10.73933745385851493502417177774

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