↑

Properties

Label 2-700-100.23-c1-0-29
Degree $2$
Conductor $700$
Sign $0.994 + 0.105i$
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  + (−1.01 − 0.981i)2-s + (−0.0725 − 0.142i)3-s + (0.0719 + 1.99i)4-s + (1.73 + 1.41i)5-s + (−0.0659 + 0.216i)6-s + (0.707 + 0.707i)7-s + (1.88 − 2.10i)8-s + (1.74 − 2.40i)9-s + (−0.377 − 3.13i)10-s + (0.485 + 0.667i)11-s + (0.279 − 0.155i)12-s + (0.338 + 2.13i)13-s + (−0.0254 − 1.41i)14-s + (0.0753 − 0.349i)15-s + (−3.98 + 0.287i)16-s + (−0.484 − 0.246i)17-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s + (−0.0418 − 0.0822i)3-s + (0.0359 + 0.999i)4-s + (0.775 + 0.631i)5-s + (−0.0269 + 0.0882i)6-s + (0.267 + 0.267i)7-s + (0.667 − 0.744i)8-s + (0.582 − 0.802i)9-s + (−0.119 − 0.992i)10-s + (0.146 + 0.201i)11-s + (0.0806 − 0.0448i)12-s + (0.0939 + 0.593i)13-s + (−0.00679 − 0.377i)14-s + (0.0194 − 0.0902i)15-s + (−0.997 + 0.0719i)16-s + (−0.117 − 0.0598i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.994 + 0.105i$
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: \((2,\ 700,\ (\ :1/2),\ 0.994 + 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28767 - 0.0684110i\)
\(L(\frac12)\) \(\approx\) \(1.28767 - 0.0684110i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.01 + 0.981i)T \)
5 \( 1 + (-1.73 - 1.41i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.0725 + 0.142i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-0.485 - 0.667i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.338 - 2.13i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (0.484 + 0.246i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.598 + 1.84i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.531 - 3.35i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-5.91 - 1.92i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.81 + 1.56i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.61 - 0.414i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-3.28 - 2.38i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.18 - 4.18i)T - 43iT^{2} \)
47 \( 1 + (-7.12 + 3.62i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (7.96 - 4.06i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (3.25 + 2.36i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-6.77 + 4.92i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-4.11 + 8.07i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-12.7 - 4.14i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.91 + 0.936i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-1.57 + 4.85i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.70 - 2.39i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-2.53 - 3.49i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.0312 - 0.0612i)T + (-57.0 + 78.4i)T^{2} \)
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

−10.33290350619812480511730238302, −9.574472671746255124662360229599, −9.072574987676930606539911974085, −7.951302539279558628269503274803, −6.88771396317748167497354350013, −6.35224948791412892212441513211, −4.78538307604044765339300055053, −3.59362316324133976199687397495, −2.45743278366789841490589116274, −1.34265181086430261523225039284, 1.04720260333871962777434407253, 2.27665880849305697792250694207, 4.38628435192574111314743677002, 5.17506343183265750517493930820, 6.06760881872276725464441020893, 6.98767847009686753462529283999, 8.101169104896624728057547393275, 8.532636976462651672693069567695, 9.636318760597878923200957574060, 10.31547107614807682737732319550

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