↑

Properties

Label 2-850-85.78-c1-0-18
Degree $2$
Conductor $850$
Sign $0.767 + 0.641i$
Analytic cond. $6.78728$
Root an. cond. $2.60524$
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  + (0.382 + 0.923i)2-s + (−0.442 − 2.22i)3-s + (−0.707 + 0.707i)4-s + (1.88 − 1.26i)6-s + (1.93 − 1.29i)7-s + (−0.923 − 0.382i)8-s + (−1.98 + 0.820i)9-s + (1.86 + 2.78i)11-s + (1.88 + 1.26i)12-s + 4.73·13-s + (1.93 + 1.29i)14-s − i·16-s + (−3.36 − 2.38i)17-s + (−1.51 − 1.51i)18-s + (0.786 + 0.325i)19-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (−0.255 − 1.28i)3-s + (−0.353 + 0.353i)4-s + (0.769 − 0.514i)6-s + (0.731 − 0.488i)7-s + (−0.326 − 0.135i)8-s + (−0.660 + 0.273i)9-s + (0.561 + 0.840i)11-s + (0.544 + 0.363i)12-s + 1.31·13-s + (0.517 + 0.345i)14-s − 0.250i·16-s + (−0.815 − 0.578i)17-s + (−0.357 − 0.357i)18-s + (0.180 + 0.0747i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.767 + 0.641i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ 0.767 + 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67824 - 0.608860i\)
\(L(\frac12)\) \(\approx\) \(1.67824 - 0.608860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 0.923i)T \)
5 \( 1 \)
17 \( 1 + (3.36 + 2.38i)T \)
good3 \( 1 + (0.442 + 2.22i)T + (-2.77 + 1.14i)T^{2} \)
7 \( 1 + (-1.93 + 1.29i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-1.86 - 2.78i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 - 4.73T + 13T^{2} \)
19 \( 1 + (-0.786 - 0.325i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.12 + 0.820i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (-3.83 + 0.762i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + (-4.45 + 6.66i)T + (-11.8 - 28.6i)T^{2} \)
37 \( 1 + (-9.93 + 1.97i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (0.472 + 0.0940i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (0.702 - 1.69i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 6.52iT - 47T^{2} \)
53 \( 1 + (3.22 - 1.33i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (5.52 + 13.3i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.17 - 5.90i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-0.912 - 0.912i)T + 67iT^{2} \)
71 \( 1 + (-10.9 - 7.33i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (10.1 + 6.77i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-6.77 + 4.52i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (3.72 + 8.98i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (2.75 - 2.75i)T - 89iT^{2} \)
97 \( 1 + (-7.64 - 5.10i)T + (37.1 + 89.6i)T^{2} \)
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

−10.03197019496169168831845041946, −8.978908719075742508574841460883, −7.954566712931030284724226822414, −7.57548082999713028868038222777, −6.46903305393840917647153188350, −6.20806897414937869580701338355, −4.76045124175921427944932153392, −3.97416595797263946294554743314, −2.20099098000632130502861244690, −0.977764236950555286324649862660, 1.43670839602344374373775648483, 3.04267092922784627763575169652, 4.02353850746911936133837885265, 4.65633228124694624887690418604, 5.70414260795856169417207432257, 6.39657759614317088863934157769, 8.234218807928529247506070614454, 8.735599649689829582329450107801, 9.580161632461479076186701689807, 10.47886307710487813061876860880

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