Properties

Label 2-91-91.24-c1-0-3
Degree $2$
Conductor $91$
Sign $0.698 + 0.716i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.38 + 0.638i)2-s − 0.168i·3-s + (3.53 − 2.04i)4-s + (−2.38 − 0.638i)5-s + (0.107 + 0.402i)6-s + (0.932 − 2.47i)7-s + (−3.63 + 3.63i)8-s + 2.97·9-s + 6.08·10-s + (3.22 − 3.22i)11-s + (−0.344 − 0.596i)12-s + (−1.54 − 3.25i)13-s + (−0.640 + 6.49i)14-s + (−0.107 + 0.402i)15-s + (2.24 − 3.89i)16-s + (0.0563 + 0.0976i)17-s + ⋯
L(s)  = 1  + (−1.68 + 0.451i)2-s − 0.0974i·3-s + (1.76 − 1.02i)4-s + (−1.06 − 0.285i)5-s + (0.0440 + 0.164i)6-s + (0.352 − 0.935i)7-s + (−1.28 + 1.28i)8-s + 0.990·9-s + 1.92·10-s + (0.971 − 0.971i)11-s + (−0.0994 − 0.172i)12-s + (−0.429 − 0.903i)13-s + (−0.171 + 1.73i)14-s + (−0.0278 + 0.103i)15-s + (0.562 − 0.973i)16-s + (0.0136 + 0.0236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.698 + 0.716i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.698 + 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.390863 - 0.164810i\)
\(L(\frac12)\) \(\approx\) \(0.390863 - 0.164810i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.932 + 2.47i)T \)
13 \( 1 + (1.54 + 3.25i)T \)
good2 \( 1 + (2.38 - 0.638i)T + (1.73 - i)T^{2} \)
3 \( 1 + 0.168iT - 3T^{2} \)
5 \( 1 + (2.38 + 0.638i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.22 + 3.22i)T - 11iT^{2} \)
17 \( 1 + (-0.0563 - 0.0976i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.43 - 2.43i)T - 19iT^{2} \)
23 \( 1 + (0.565 + 0.326i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.82 + 4.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.43 - 5.34i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.402 - 1.50i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-10.6 - 2.86i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.08 - 3.51i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.53 - 5.72i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.41 - 4.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.02 - 3.81i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 15.3iT - 61T^{2} \)
67 \( 1 + (-4.44 - 4.44i)T + 67iT^{2} \)
71 \( 1 + (-3.56 + 0.955i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.43 - 0.651i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.11 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.34 + 3.34i)T - 83iT^{2} \)
89 \( 1 + (8.39 - 2.24i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.04 + 3.89i)T + (-84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.30646831276227374891242981428, −12.66927063972594495124329631046, −11.36431635690902685216632639826, −10.54049609395224742395489937397, −9.448557579093130181043671984139, −8.087361596466150580271642796407, −7.66571926134659644263426643004, −6.41842868236851010421471490291, −4.10147515153589279188533410942, −0.966401762260984087081071375357, 2.03385495081722024365230972287, 4.22478416401697123989447413091, 6.89084035265642284133324861548, 7.63250965647724735298910778969, 8.961695455967939141949257133076, 9.619125011923384587012747218726, 10.96098592093275836261608215316, 11.78712940084683705212043738247, 12.47572163437911232228903196245, 14.72217786414251000528770763339

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