↑

Properties

Label 2-700-100.23-c1-0-12
Degree $2$
Conductor $700$
Sign $0.709 - 0.704i$
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  + (−0.126 + 1.40i)2-s + (−1.45 − 2.86i)3-s + (−1.96 − 0.356i)4-s + (1.35 + 1.77i)5-s + (4.21 − 1.69i)6-s + (−0.707 − 0.707i)7-s + (0.750 − 2.72i)8-s + (−4.30 + 5.92i)9-s + (−2.67 + 1.68i)10-s + (2.04 + 2.81i)11-s + (1.85 + 6.15i)12-s + (0.0115 + 0.0730i)13-s + (1.08 − 0.906i)14-s + (3.10 − 6.48i)15-s + (3.74 + 1.40i)16-s + (−2.40 − 1.22i)17-s + ⋯
L(s)  = 1  + (−0.0894 + 0.995i)2-s + (−0.842 − 1.65i)3-s + (−0.984 − 0.178i)4-s + (0.607 + 0.794i)5-s + (1.72 − 0.690i)6-s + (−0.267 − 0.267i)7-s + (0.265 − 0.964i)8-s + (−1.43 + 1.97i)9-s + (−0.845 + 0.534i)10-s + (0.616 + 0.848i)11-s + (0.534 + 1.77i)12-s + (0.00320 + 0.0202i)13-s + (0.290 − 0.242i)14-s + (0.800 − 1.67i)15-s + (0.936 + 0.350i)16-s + (−0.584 − 0.297i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.709 - 0.704i$
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: \((2,\ 700,\ (\ :1/2),\ 0.709 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895759 + 0.369219i\)
\(L(\frac12)\) \(\approx\) \(0.895759 + 0.369219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.126 - 1.40i)T \)
5 \( 1 + (-1.35 - 1.77i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.45 + 2.86i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-2.04 - 2.81i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.0115 - 0.0730i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (2.40 + 1.22i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.280 + 0.863i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.183 - 1.15i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-7.67 - 2.49i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.36 + 1.74i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.47 + 0.549i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-8.36 - 6.08i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.55 - 4.55i)T - 43iT^{2} \)
47 \( 1 + (-11.8 + 6.01i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-9.84 + 5.01i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-1.40 - 1.02i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.22 - 2.33i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.00 - 5.90i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (12.5 + 4.08i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-10.3 - 1.63i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (3.55 - 10.9i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.82 + 1.94i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-0.877 - 1.20i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (6.30 + 12.3i)T + (-57.0 + 78.4i)T^{2} \)
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

−10.51173283350116246490870762461, −9.659494902612429588273681243862, −8.519287699648010834031193473560, −7.45709194965945978212499870379, −6.91529823812725119088945227637, −6.41771390723454309887885544486, −5.66474230906851126546320780290, −4.48764850585535093382936240063, −2.54596956261587989834614616418, −1.09828153339349394178081759547, 0.75177304842007798316434244665, 2.75411212924253100953363566401, 4.03106680150798914260993027416, 4.56959986064602142142039591317, 5.62152608541158533092758370864, 6.19695072884677701588388806174, 8.495145882252110355088083129027, 8.961072230717027676725289170552, 9.653621953449913790920699842800, 10.40940749324150402185607121787

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