↑

Properties

Label 2-91-13.3-c1-0-0
Degree $2$
Conductor $91$
Sign $0.859 + 0.511i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
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  + (−1.30 − 2.26i)2-s + (1.30 + 2.26i)3-s + (−2.42 + 4.20i)4-s + 2.61·5-s + (3.42 − 5.93i)6-s + (−0.5 + 0.866i)7-s + 7.47·8-s + (−1.92 + 3.33i)9-s + (−3.42 − 5.93i)10-s + (−0.927 − 1.60i)11-s − 12.7·12-s + (−2.5 − 2.59i)13-s + 2.61·14-s + (3.42 + 5.93i)15-s + (−4.92 − 8.53i)16-s + (0.736 − 1.27i)17-s + ⋯
L(s)  = 1  + (−0.925 − 1.60i)2-s + (0.755 + 1.30i)3-s + (−1.21 + 2.10i)4-s + 1.17·5-s + (1.39 − 2.42i)6-s + (−0.188 + 0.327i)7-s + 2.64·8-s + (−0.642 + 1.11i)9-s + (−1.08 − 1.87i)10-s + (−0.279 − 0.484i)11-s − 3.66·12-s + (−0.693 − 0.720i)13-s + 0.699·14-s + (0.884 + 1.53i)15-s + (−1.23 − 2.13i)16-s + (0.178 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.801490 - 0.220276i\)
\(L(\frac12)\) \(\approx\) \(0.801490 - 0.220276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (1.30 + 2.26i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.30 - 2.26i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.61T + 5T^{2} \)
11 \( 1 + (0.927 + 1.60i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.736 + 1.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.927 + 1.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.54 + 6.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.70T + 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.381 + 0.661i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.28 - 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.23T + 47T^{2} \)
53 \( 1 - 3.76T + 53T^{2} \)
59 \( 1 + (-1.11 + 1.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.35 - 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.09 + 12.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + (2.45 + 4.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.42 - 16.3i)T + (-48.5 - 84.0i)T^{2} \)
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

−13.69585316563607396005583691059, −12.88742228761098839182673088396, −11.41799207551908262378884258618, −10.40739824166229282417700423445, −9.607390863362311771317061218147, −9.265014826457146716128406306395, −8.006658353264053116592906854008, −5.21395122041646884883505643151, −3.47108356443198913441640188356, −2.42080627102110695534675623945, 1.80214490826245528977902820077, 5.36404778182236042436944321575, 6.70797965088300389274497248723, 7.20714144755062709813776420676, 8.402632954584516310066139809028, 9.348664020825086494977289568415, 10.27514006967594240862702061569, 12.61119931919059129964950553941, 13.60944668620195947049057451698, 14.27003126846218231628705924284

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