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Properties

Label 2-700-140.19-c1-0-51
Degree $2$
Conductor $700$
Sign $-0.389 + 0.920i$
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 + (1.5 − 0.866i)3-s + (−1.73 + i)4-s + (−1.73 − 1.73i)6-s + (2 − 1.73i)7-s + (2 + 1.99i)8-s + (0.866 − 0.5i)11-s + (−1.73 + 3i)12-s + 3.46·13-s + (−3.09 − 2.09i)14-s + (1.99 − 3.46i)16-s + (−0.866 − 1.5i)17-s + (2.59 − 4.5i)19-s + (1.50 − 4.33i)21-s + (−1 − 0.999i)22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.866 − 0.499i)3-s + (−0.866 + 0.5i)4-s + (−0.707 − 0.707i)6-s + (0.755 − 0.654i)7-s + (0.707 + 0.707i)8-s + (0.261 − 0.150i)11-s + (−0.499 + 0.866i)12-s + 0.960·13-s + (−0.827 − 0.560i)14-s + (0.499 − 0.866i)16-s + (−0.210 − 0.363i)17-s + (0.596 − 1.03i)19-s + (0.327 − 0.944i)21-s + (−0.213 − 0.213i)22-s + (0.104 − 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.389 + 0.920i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.389 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985908 - 1.48768i\)
\(L(\frac12)\) \(\approx\) \(0.985908 - 1.48768i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good3 \( 1 + (-1.5 + 0.866i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + (0.866 + 1.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.59 + 4.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (0.866 + 1.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.59 + 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (-7.5 - 4.33i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.59 - 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14iT - 71T^{2} \)
73 \( 1 + (4.33 + 7.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.79 + 4.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + (13.5 + 7.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.3T + 97T^{2} \)
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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

−10.29810953477229359713692446140, −9.088509394964883773528698274024, −8.690398869103533851808974031846, −7.74538544110410122402442366655, −7.11173847610861189008707184930, −5.42407913160834540128763876919, −4.30404233087412887125534308268, −3.32184001676576729709230177566, −2.22036206220545976712334506146, −1.08973939949667980379347666164, 1.63615869036582801559625912839, 3.40627642382740140959326295056, 4.29793959054472569820814567109, 5.47367602337224744985920401331, 6.21675500437940924871750369188, 7.45856296287991111435719889427, 8.329238989130423596610684774755, 8.784390268790069002135213989887, 9.522606076433167199569821060406, 10.39450376470148303005216250693

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