↑

Properties

Label 2-700-7.2-c1-0-12
Degree $2$
Conductor $700$
Sign $-0.850 - 0.526i$
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  + (−1.60 − 2.77i)3-s + (1.53 − 2.15i)7-s + (−3.63 + 6.29i)9-s + (−2.10 − 3.64i)11-s − 0.204·13-s + (−2.53 − 4.38i)17-s + (−0.531 + 0.921i)19-s + (−8.44 − 0.794i)21-s + (1.07 − 1.85i)23-s + 13.6·27-s − 7.47·29-s + (4.23 + 7.33i)31-s + (−6.73 + 11.6i)33-s + (−5.30 + 9.19i)37-s + (0.327 + 0.567i)39-s + ⋯
L(s)  = 1  + (−0.925 − 1.60i)3-s + (0.579 − 0.815i)7-s + (−1.21 + 2.09i)9-s + (−0.633 − 1.09i)11-s − 0.0566·13-s + (−0.614 − 1.06i)17-s + (−0.122 + 0.211i)19-s + (−1.84 − 0.173i)21-s + (0.223 − 0.386i)23-s + 2.63·27-s − 1.38·29-s + (0.760 + 1.31i)31-s + (−1.17 + 2.03i)33-s + (−0.872 + 1.51i)37-s + (0.0524 + 0.0908i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.850 - 0.526i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.850 - 0.526i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166151 + 0.584262i\)
\(L(\frac12)\) \(\approx\) \(0.166151 + 0.584262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-1.53 + 2.15i)T \)
good3 \( 1 + (1.60 + 2.77i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.10 + 3.64i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.204T + 13T^{2} \)
17 \( 1 + (2.53 + 4.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.531 - 0.921i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.07 + 1.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.47T + 29T^{2} \)
31 \( 1 + (-4.23 - 7.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.30 - 9.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 8.26T + 43T^{2} \)
47 \( 1 + (-1.63 + 2.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.83 - 4.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.602 + 1.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.827 - 1.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.20 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.591T + 71T^{2} \)
73 \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.27 - 5.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.88T + 83T^{2} \)
89 \( 1 + (-4.63 + 8.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.33T + 97T^{2} \)
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

−10.34807729060186338170044782966, −8.740917924576914870139985801181, −7.958817549138886817162243340399, −7.22859523033925875667345348692, −6.53734652845685562794983717612, −5.54755018105305514980427042137, −4.73906316095598428498766840137, −2.93497072684422449293185338123, −1.55311265507720330531391621592, −0.35591637196293951401053308229, 2.26887021368931969935778003362, 3.86357547417262783338367920911, 4.62751000845830273907198144500, 5.44417214543593119033217151039, 6.10509129575648807366596964926, 7.49180901607581142339207563840, 8.700324537946356191384424839571, 9.408629758809970842924295489114, 10.15168705109443990566066155938, 10.94416583451034691448625799718

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