Properties

Label 2-700-140.103-c0-0-0
Degree $2$
Conductor $700$
Sign $0.350 - 0.936i$
Analytic cond. $0.349345$
Root an. cond. $0.591054$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.448 + 1.67i)3-s + (0.866 − 0.499i)4-s + 1.73i·6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (−1.73 − 1.00i)9-s + (0.448 + 1.67i)12-s + i·14-s + (0.500 − 0.866i)16-s + (−1.93 − 0.517i)18-s + (−1.50 − 0.866i)21-s + (−0.258 − 0.965i)23-s + (0.866 + 1.5i)24-s + (1.22 − 1.22i)27-s + (0.258 + 0.965i)28-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.448 + 1.67i)3-s + (0.866 − 0.499i)4-s + 1.73i·6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (−1.73 − 1.00i)9-s + (0.448 + 1.67i)12-s + i·14-s + (0.500 − 0.866i)16-s + (−1.93 − 0.517i)18-s + (−1.50 − 0.866i)21-s + (−0.258 − 0.965i)23-s + (0.866 + 1.5i)24-s + (1.22 − 1.22i)27-s + (0.258 + 0.965i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.350 - 0.936i$
Analytic conductor: \(0.349345\)
Root analytic conductor: \(0.591054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :0),\ 0.350 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.394685283\)
\(L(\frac12)\) \(\approx\) \(1.394685283\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (0.258 - 0.965i)T \)
good3 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
89 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.72402301863372233970943256509, −10.28607976040705329089981407997, −9.337821263478711338493481284866, −8.589581196910798735308856067003, −6.93947372539664195270563660466, −5.83718016116654840148847588693, −5.34109147541218438055301842866, −4.40510366695514855028385869107, −3.54966038497482328875417548074, −2.49328113025985959878365685198, 1.41964317179010972080787827297, 2.73088878189420697766663729175, 4.02795197713944862910319567984, 5.29148411484387431366615563296, 6.25559622452298971139573877461, 6.78192031468000646887469572965, 7.67590624654212101181122828828, 8.083113296062513697235440927042, 9.765077021344419031182412793342, 11.02957297752899223526044502221

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