↑

Properties

Label 2-875-175.17-c1-0-6
Degree $2$
Conductor $875$
Sign $-0.383 - 0.923i$
Analytic cond. $6.98691$
Root an. cond. $2.64327$
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  + (−0.0171 + 0.000898i)2-s + (0.711 − 0.879i)3-s + (−1.98 + 0.209i)4-s + (−0.0114 + 0.0157i)6-s + (−0.822 − 2.51i)7-s + (0.0678 − 0.0107i)8-s + (0.357 + 1.68i)9-s + (−4.96 − 1.05i)11-s + (−1.23 + 1.89i)12-s + (0.286 − 0.145i)13-s + (0.0163 + 0.0423i)14-s + (3.91 − 0.831i)16-s + (−1.33 + 3.48i)17-s + (−0.00764 − 0.0285i)18-s + (−0.698 + 6.64i)19-s + ⋯
L(s)  = 1  + (−0.0121 + 0.000635i)2-s + (0.410 − 0.507i)3-s + (−0.994 + 0.104i)4-s + (−0.00466 + 0.00641i)6-s + (−0.310 − 0.950i)7-s + (0.0239 − 0.00379i)8-s + (0.119 + 0.561i)9-s + (−1.49 − 0.317i)11-s + (−0.355 + 0.547i)12-s + (0.0793 − 0.0404i)13-s + (0.00437 + 0.0113i)14-s + (0.977 − 0.207i)16-s + (−0.324 + 0.846i)17-s + (−0.00180 − 0.00672i)18-s + (−0.160 + 1.52i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $-0.383 - 0.923i$
Analytic conductor: \(6.98691\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :1/2),\ -0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242709 + 0.363406i\)
\(L(\frac12)\) \(\approx\) \(0.242709 + 0.363406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (0.822 + 2.51i)T \)
good2 \( 1 + (0.0171 - 0.000898i)T + (1.98 - 0.209i)T^{2} \)
3 \( 1 + (-0.711 + 0.879i)T + (-0.623 - 2.93i)T^{2} \)
11 \( 1 + (4.96 + 1.05i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.286 + 0.145i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (1.33 - 3.48i)T + (-12.6 - 11.3i)T^{2} \)
19 \( 1 + (0.698 - 6.64i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (-0.178 - 3.41i)T + (-22.8 + 2.40i)T^{2} \)
29 \( 1 + (1.99 + 2.74i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.305 - 0.686i)T + (-20.7 + 23.0i)T^{2} \)
37 \( 1 + (-2.76 - 1.79i)T + (15.0 + 33.8i)T^{2} \)
41 \( 1 + (7.88 + 2.56i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (3.99 - 3.99i)T - 43iT^{2} \)
47 \( 1 + (5.87 - 2.25i)T + (34.9 - 31.4i)T^{2} \)
53 \( 1 + (-8.77 - 7.10i)T + (11.0 + 51.8i)T^{2} \)
59 \( 1 + (6.39 + 7.10i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (2.03 + 1.83i)T + (6.37 + 60.6i)T^{2} \)
67 \( 1 + (6.09 + 2.33i)T + (49.7 + 44.8i)T^{2} \)
71 \( 1 + (3.66 - 2.66i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.50 - 8.47i)T + (-29.6 + 66.6i)T^{2} \)
79 \( 1 + (3.33 - 7.49i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (0.203 + 1.28i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-0.663 + 0.736i)T + (-9.30 - 88.5i)T^{2} \)
97 \( 1 + (1.94 - 12.2i)T + (-92.2 - 29.9i)T^{2} \)
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

−10.29887234853808442064079359317, −9.710494720369658120259210821282, −8.330977498881474353584307194832, −8.077730826292608013955728007123, −7.31046078349309596176997042516, −6.01070834525379199034765061688, −5.06314565829038864640853871341, −4.04784426800463674994466598125, −3.10041524629567193847474031786, −1.56585454818402770627971183397, 0.20422104876984762973292017318, 2.49719503969884192782218677969, 3.36750430635611804827651623006, 4.72039799609550020446537907044, 5.11991131823579913345081383984, 6.33574209134262075311002449291, 7.44931144534562879531540177452, 8.732145601886037921612553163538, 8.849712628838249160083100537796, 9.815903368897382596209095563658

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