↑

Properties

Label 2-95-19.11-c1-0-2
Degree $2$
Conductor $95$
Sign $0.689 - 0.724i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + (0.595 − 1.03i)2-s + (−1.52 + 2.63i)3-s + (0.290 + 0.503i)4-s + (−0.5 + 0.866i)5-s + (1.81 + 3.14i)6-s − 0.609·7-s + 3.07·8-s + (−3.14 − 5.44i)9-s + (0.595 + 1.03i)10-s + 4.48·11-s − 1.77·12-s + (−2.21 − 3.84i)13-s + (−0.362 + 0.628i)14-s + (−1.52 − 2.63i)15-s + (1.24 − 2.16i)16-s + (−1.45 + 2.51i)17-s + ⋯
L(s)  = 1  + (0.421 − 0.729i)2-s + (−0.879 + 1.52i)3-s + (0.145 + 0.251i)4-s + (−0.223 + 0.387i)5-s + (0.740 + 1.28i)6-s − 0.230·7-s + 1.08·8-s + (−1.04 − 1.81i)9-s + (0.188 + 0.326i)10-s + 1.35·11-s − 0.511·12-s + (−0.615 − 1.06i)13-s + (−0.0969 + 0.167i)14-s + (−0.393 − 0.681i)15-s + (0.312 − 0.540i)16-s + (−0.352 + 0.609i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.689 - 0.724i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.689 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923333 + 0.395663i\)
\(L(\frac12)\) \(\approx\) \(0.923333 + 0.395663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-3.60 + 2.44i)T \)
good2 \( 1 + (-0.595 + 1.03i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.52 - 2.63i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 0.609T + 7T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
13 \( 1 + (2.21 + 3.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.45 - 2.51i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.42 - 2.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.558 + 0.966i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.22T + 31T^{2} \)
37 \( 1 + 3.77T + 37T^{2} \)
41 \( 1 + (-4.15 + 7.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.99 + 8.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.94 - 5.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.22 + 7.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.11 - 8.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.49 - 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.23 + 7.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.80 - 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.86 - 3.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.51 - 7.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.12T + 83T^{2} \)
89 \( 1 + (3.96 + 6.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.83 + 8.37i)T + (-48.5 - 84.0i)T^{2} \)
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

−14.23868012159165376631846781055, −12.67391077390494367521267730264, −11.74294136291820441402249928482, −11.02955804851961730728192127046, −10.21771716535065410158016132275, −9.121320607968676259237701201843, −7.17837947516048600967292796279, −5.58188531409221067478323579495, −4.21826795674787904797372144880, −3.30775523632164727215163005491, 1.52314459379814844292666739878, 4.73530232155458301232763742918, 6.05154862647767140795018180846, 6.83539010295365619430300297276, 7.61112601488613443735634562951, 9.335171142127282285750283512676, 11.16258517043112513025399650256, 11.89601514793113129694594310482, 12.81931980662551357343448134467, 13.93157707571561006124095143163

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