Properties

Label 2-700-140.139-c1-0-0
Degree $2$
Conductor $700$
Sign $-0.621 + 0.783i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.17 − 0.780i)2-s + 3.02i·3-s + (0.780 + 1.84i)4-s + (2.35 − 3.56i)6-s + (−1.51 + 2.17i)7-s + (0.516 − 2.78i)8-s − 6.12·9-s − 4.71i·11-s + (−5.56 + 2.35i)12-s − 2·13-s + (3.47 − 1.38i)14-s + (−2.78 + 2.87i)16-s + 1.12·17-s + (7.22 + 4.78i)18-s − 4.71·19-s + ⋯
L(s)  = 1  + (−0.833 − 0.552i)2-s + 1.74i·3-s + (0.390 + 0.920i)4-s + (0.962 − 1.45i)6-s + (−0.570 + 0.821i)7-s + (0.182 − 0.983i)8-s − 2.04·9-s − 1.42i·11-s + (−1.60 + 0.680i)12-s − 0.554·13-s + (0.929 − 0.369i)14-s + (−0.695 + 0.718i)16-s + 0.272·17-s + (1.70 + 1.12i)18-s − 1.08·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.621 + 0.783i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0452312 - 0.0935511i\)
\(L(\frac12)\) \(\approx\) \(0.0452312 - 0.0935511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.780i)T \)
5 \( 1 \)
7 \( 1 + (1.51 - 2.17i)T \)
good3 \( 1 - 3.02iT - 3T^{2} \)
11 \( 1 + 4.71iT - 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 1.12T + 17T^{2} \)
19 \( 1 + 4.71T + 19T^{2} \)
23 \( 1 + 6.41T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 3.39T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 1.12iT - 41T^{2} \)
43 \( 1 - 0.371T + 43T^{2} \)
47 \( 1 + 5.08iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 2.06T + 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 3.76T + 67T^{2} \)
71 \( 1 - 7.36iT - 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 - 1.32iT - 79T^{2} \)
83 \( 1 - 3.02iT - 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 + 1.12T + 97T^{2} \)
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

−10.68351684852715787490114782548, −10.10877648591981836478293804540, −9.450620554004941726437651675147, −8.644733272977743583415997670440, −8.184153338456234790973946429364, −6.44454171148665628066606258149, −5.58878584716175266473338099472, −4.29559347447090011779615536570, −3.38938511181463043034400568312, −2.55407411837486769237829937892, 0.06850876168250010517407762868, 1.54294458480903575349032773132, 2.50500809853489120861107401506, 4.52133992650046334350965820473, 6.01479470136946885971160862097, 6.59902732716320833159022609329, 7.42669783790062055876953309022, 7.73972189612144339172460413673, 8.806728194356495658509037160992, 9.888001989663708168339868074307

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