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Properties

Label 2-700-28.19-c1-0-11
Degree $2$
Conductor $700$
Sign $-0.614 + 0.788i$
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  + (−0.569 + 1.29i)2-s + (1.51 + 2.62i)3-s + (−1.35 − 1.47i)4-s + (−4.25 + 0.465i)6-s + (−2.57 − 0.602i)7-s + (2.67 − 0.908i)8-s + (−3.08 + 5.33i)9-s + (−1.03 + 0.598i)11-s + (1.82 − 5.77i)12-s + 4.83i·13-s + (2.24 − 2.99i)14-s + (−0.349 + 3.98i)16-s + (−2.20 + 1.27i)17-s + (−5.15 − 7.02i)18-s + (0.711 − 1.23i)19-s + ⋯
L(s)  = 1  + (−0.402 + 0.915i)2-s + (0.873 + 1.51i)3-s + (−0.675 − 0.737i)4-s + (−1.73 + 0.190i)6-s + (−0.973 − 0.227i)7-s + (0.946 − 0.321i)8-s + (−1.02 + 1.77i)9-s + (−0.312 + 0.180i)11-s + (0.525 − 1.66i)12-s + 1.34i·13-s + (0.600 − 0.799i)14-s + (−0.0873 + 0.996i)16-s + (−0.534 + 0.308i)17-s + (−1.21 − 1.65i)18-s + (0.163 − 0.282i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.614 + 0.788i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.614 + 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382022 - 0.782203i\)
\(L(\frac12)\) \(\approx\) \(0.382022 - 0.782203i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.569 - 1.29i)T \)
5 \( 1 \)
7 \( 1 + (2.57 + 0.602i)T \)
good3 \( 1 + (-1.51 - 2.62i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.03 - 0.598i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.83iT - 13T^{2} \)
17 \( 1 + (2.20 - 1.27i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.711 + 1.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.02 + 2.90i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.774T + 29T^{2} \)
31 \( 1 + (3.31 + 5.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.55 + 4.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.46iT - 41T^{2} \)
43 \( 1 - 1.38iT - 43T^{2} \)
47 \( 1 + (-0.535 + 0.927i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.68 - 2.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.94 - 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.31 - 4.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.14 - 5.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 16.3iT - 71T^{2} \)
73 \( 1 + (0.0927 - 0.0535i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.32 + 5.38i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + (3.41 + 1.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.71iT - 97T^{2} \)
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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

−10.43265936340075645612917318602, −9.909710151926816804882544908888, −9.221198811517148732375754016325, −8.703700990271119237390897707321, −7.67662456641469550173178153635, −6.65166719918301299436218648955, −5.65439689843564083684802599268, −4.34896709510052637056794682111, −4.01758112990438524699708740074, −2.45218836886539329658567164860, 0.44935652578146817318926862882, 1.92434191703245735333548079448, 2.91825385688003622340033198172, 3.56836554748383799996661404245, 5.46793762563879348059297819629, 6.63329938220580356415099973208, 7.57164380458885044289857793900, 8.198752302041021999809221402427, 8.960899844930509936014989385568, 9.805834504073095492155394597772

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