↑

Properties

Label 2-700-100.23-c1-0-22
Degree $2$
Conductor $700$
Sign $0.846 - 0.531i$
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  + (0.718 − 1.21i)2-s + (0.713 + 1.40i)3-s + (−0.968 − 1.74i)4-s + (−1.23 + 1.86i)5-s + (2.21 + 0.136i)6-s + (0.707 + 0.707i)7-s + (−2.82 − 0.0760i)8-s + (0.311 − 0.429i)9-s + (1.38 + 2.84i)10-s + (0.0883 + 0.121i)11-s + (1.75 − 2.60i)12-s + (0.899 + 5.67i)13-s + (1.36 − 0.353i)14-s + (−3.49 − 0.399i)15-s + (−2.12 + 3.39i)16-s + (4.70 + 2.39i)17-s + ⋯
L(s)  = 1  + (0.507 − 0.861i)2-s + (0.411 + 0.808i)3-s + (−0.484 − 0.874i)4-s + (−0.552 + 0.833i)5-s + (0.905 + 0.0556i)6-s + (0.267 + 0.267i)7-s + (−0.999 − 0.0268i)8-s + (0.103 − 0.143i)9-s + (0.437 + 0.899i)10-s + (0.0266 + 0.0366i)11-s + (0.507 − 0.751i)12-s + (0.249 + 1.57i)13-s + (0.365 − 0.0945i)14-s + (−0.901 − 0.103i)15-s + (−0.530 + 0.847i)16-s + (1.14 + 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.846 - 0.531i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.846 - 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85129 + 0.532930i\)
\(L(\frac12)\) \(\approx\) \(1.85129 + 0.532930i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.718 + 1.21i)T \)
5 \( 1 + (1.23 - 1.86i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.713 - 1.40i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-0.0883 - 0.121i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.899 - 5.67i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-4.70 - 2.39i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.209 - 0.644i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.420 - 2.65i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-0.763 - 0.248i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.71 - 2.18i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.42 + 0.542i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-4.00 - 2.90i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.01 + 3.01i)T - 43iT^{2} \)
47 \( 1 + (5.73 - 2.91i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-0.0497 + 0.0253i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (8.65 + 6.28i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-12.2 + 8.91i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.58 + 7.04i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (6.06 + 1.97i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (9.06 + 1.43i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-0.750 + 2.31i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-6.15 - 3.13i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (2.62 + 3.61i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.186 - 0.366i)T + (-57.0 + 78.4i)T^{2} \)
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

−10.65101371728161353000882351886, −9.717934910011696410793480612114, −9.218829871531590040033409376198, −8.131436440457024529245495744824, −6.88078029503486705825455249098, −5.89412941170476207777971744660, −4.61887780963957025266310073426, −3.83322250712441979908410914938, −3.18034726584197172087933795415, −1.75817752572539118556951654016, 0.895307718317830924129676524561, 2.83287821892620970640358042929, 3.98359002447762892193591572731, 5.06579681606488429056980808833, 5.79204338838302364565577612897, 7.15805482959391892612704431149, 7.76323418389282025582560090642, 8.188969728579682976033453983045, 9.100980412073511675267059272681, 10.30587475845108418101916975227

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