Properties

Label 1-91-91.16-r0-0-0
Degree $1$
Conductor $91$
Sign $0.325 - 0.945i$
Analytic cond. $0.422602$
Root an. cond. $0.422602$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)15-s + 16-s + 17-s + (−0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + ⋯
L(s)  = 1  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)15-s + 16-s + 17-s + (−0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.325 - 0.945i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ 0.325 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.148015607 - 0.8188942059i\)
\(L(\frac12)\) \(\approx\) \(1.148015607 - 0.8188942059i\)
\(L(1)\) \(\approx\) \(1.324140942 - 0.5538359036i\)
\(L(1)\) \(\approx\) \(1.324140942 - 0.5538359036i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.7352749688962183214168912400, −29.757755708472368384550495042579, −28.59154619474868728729571806893, −27.6012162402327865243824592470, −26.29002492667162924363360691024, −25.50919808371512492434275387459, −23.75904156274503170466361931295, −23.064481889964904407272401204498, −22.30903122451837835094768564156, −21.27784840915800154976349226479, −20.36258934379406551129103947414, −19.01480120652363280819690047805, −17.44363626874521811052036685939, −16.20399976267742322891809060690, −15.1343944144563409415163901230, −14.69224326813625806662058543171, −12.9980272394096008766292260042, −11.71071192622078270597637175774, −10.90025939289953996100737540494, −9.82937728956556490368271779393, −7.63344440395395449220792125942, −6.42154069036146391351012618253, −5.08826070635576404873604375864, −3.95785702087034736680176180707, −2.722101293863071919010828301635, 1.405636617404764477734188462, 3.29031156911320484484665842589, 4.98763758184693891409409141227, 5.87420098746870088055182803777, 7.33751009391216358856833881797, 8.381430361918780691701235484941, 10.68156838019510153343574421611, 11.77647998029252136527854527850, 12.65583821523367626736176633973, 13.44148487082425743935978367057, 14.7637756229022836193714376881, 16.34197866867279063176690331877, 16.77261492365786552796222764544, 18.61771021176854248056356575735, 19.59073441995526987318334539427, 20.75036921024340250118753468951, 21.81166872814354990743452588249, 23.3189513259962473894532520012, 23.5517289963570474965339744185, 24.68906067315700481723117121747, 25.420349260764780037624558447486, 27.29657764265689399405342582661, 28.55637348500254695683264622029, 29.27991385875920815603204497707, 30.228574748706561003059116558230

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