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Properties

Label 2-91-91.19-c1-0-0
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: $\chi_{91} (19, \cdot )$
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

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

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