↑

Properties

Label 2-91-91.9-c1-0-0
Degree $2$
Conductor $91$
Sign $-0.765 - 0.642i$
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  + (1.19 + 2.06i)2-s − 2.75·3-s + (−1.85 + 3.20i)4-s + (−0.491 + 0.850i)5-s + (−3.28 − 5.69i)6-s + (2.60 + 0.452i)7-s − 4.06·8-s + 4.57·9-s − 2.34·10-s − 0.587·11-s + (5.09 − 8.82i)12-s + (2.39 + 2.69i)13-s + (2.17 + 5.93i)14-s + (1.35 − 2.34i)15-s + (−1.15 − 1.99i)16-s + (3.22 − 5.58i)17-s + ⋯
L(s)  = 1  + (0.844 + 1.46i)2-s − 1.58·3-s + (−0.925 + 1.60i)4-s + (−0.219 + 0.380i)5-s + (−1.34 − 2.32i)6-s + (0.985 + 0.170i)7-s − 1.43·8-s + 1.52·9-s − 0.741·10-s − 0.177·11-s + (1.47 − 2.54i)12-s + (0.663 + 0.748i)13-s + (0.581 + 1.58i)14-s + (0.348 − 0.604i)15-s + (−0.288 − 0.498i)16-s + (0.782 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.765 - 0.642i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.765 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.325944 + 0.895261i\)
\(L(\frac12)\) \(\approx\) \(0.325944 + 0.895261i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.60 - 0.452i)T \)
13 \( 1 + (-2.39 - 2.69i)T \)
good2 \( 1 + (-1.19 - 2.06i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + 2.75T + 3T^{2} \)
5 \( 1 + (0.491 - 0.850i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 0.587T + 11T^{2} \)
17 \( 1 + (-3.22 + 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.82T + 19T^{2} \)
23 \( 1 + (4.13 + 7.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.98 + 3.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.49 - 2.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.877 + 1.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.83 - 3.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.19 + 5.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.17 + 3.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.212 + 0.368i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.00 - 5.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 2.20T + 61T^{2} \)
67 \( 1 - 7.01T + 67T^{2} \)
71 \( 1 + (1.80 + 3.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.46 + 4.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.39 - 2.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.86T + 83T^{2} \)
89 \( 1 + (-1.04 - 1.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.84 + 6.66i)T + (-48.5 + 84.0i)T^{2} \)
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

−14.60263227039845107513977765925, −13.74274332916461419166498949655, −12.36405013523884680550644687128, −11.58795783028660449075160076816, −10.53186422583079776823650743703, −8.455197160552502316210592707063, −7.17185507937392119873304782774, −6.30533611659838069061042899564, −5.24979365640528738254496654193, −4.36439657250420829042629980134, 1.34663010083685641620956609761, 3.96115763005030510270935471274, 5.07076087141044920224967328556, 5.97402045427210215135253523032, 8.118655135909821591767927136883, 10.20361135943955632463757863667, 10.78769860153146889247164446042, 11.60962026645070400238271825123, 12.37372661278975975773052845918, 13.11748812045224643934843445177

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