Properties

Label 2-95-19.7-c1-0-5
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: $\chi_{95} (26, \cdot )$
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

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

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