Properties

Label 2-700-140.139-c1-0-34
Degree $2$
Conductor $700$
Sign $0.830 - 0.556i$
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  + (0.599 + 1.28i)2-s − 0.936i·3-s + (−1.28 + 1.53i)4-s + (1.19 − 0.561i)6-s + (−0.468 − 2.60i)7-s + (−2.73 − 0.719i)8-s + 2.12·9-s + 2.39i·11-s + (1.43 + 1.19i)12-s + 2·13-s + (3.05 − 2.16i)14-s + (−0.719 − 3.93i)16-s + 7.12·17-s + (1.27 + 2.71i)18-s − 2.39·19-s + ⋯
L(s)  = 1  + (0.424 + 0.905i)2-s − 0.540i·3-s + (−0.640 + 0.768i)4-s + (0.489 − 0.229i)6-s + (−0.176 − 0.984i)7-s + (−0.967 − 0.254i)8-s + 0.707·9-s + 0.723i·11-s + (0.415 + 0.346i)12-s + 0.554·13-s + (0.816 − 0.577i)14-s + (−0.179 − 0.983i)16-s + 1.72·17-s + (0.300 + 0.640i)18-s − 0.550·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.830 - 0.556i$
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: \((2,\ 700,\ (\ :1/2),\ 0.830 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79905 + 0.546722i\)
\(L(\frac12)\) \(\approx\) \(1.79905 + 0.546722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.599 - 1.28i)T \)
5 \( 1 \)
7 \( 1 + (0.468 + 2.60i)T \)
good3 \( 1 + 0.936iT - 3T^{2} \)
11 \( 1 - 2.39iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 7.12T + 17T^{2} \)
19 \( 1 + 2.39T + 19T^{2} \)
23 \( 1 - 5.73T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 6.67T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 7.12iT - 41T^{2} \)
43 \( 1 + 7.60T + 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 6.14iT - 71T^{2} \)
73 \( 1 + 9.36T + 73T^{2} \)
79 \( 1 - 4.27iT - 79T^{2} \)
83 \( 1 + 0.936iT - 83T^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 + 7.12T + 97T^{2} \)
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

−10.24508598978273084359893787905, −9.753173803121566206999763606031, −8.413728558438311504483106777557, −7.70851546191513258179037631320, −6.94772378170229437301420233569, −6.39351272821206985877217851980, −5.08292083633555256138750149164, −4.21530023195643504068210216895, −3.17476491606323765463608195784, −1.16409457387159883069739887959, 1.27721607531681942341129047482, 2.85675365438659024879971898465, 3.63175580512690876885169275007, 4.80380557974312820641480270736, 5.60561176083287259447212288402, 6.49553972608357845787707996214, 8.076774375821704740000049946531, 8.967373134075996908462289240956, 9.648125190120449304630451016432, 10.46005694908690858923872521602

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