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Properties

Label 1-91-91.83-r0-0-0
Degree $1$
Conductor $91$
Sign $0.957 + 0.289i$
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  − i·2-s − 3-s − 4-s + i·5-s + i·6-s + i·8-s + 9-s + 10-s + i·11-s + 12-s − i·15-s + 16-s + 17-s − i·18-s + i·19-s − i·20-s + ⋯
L(s)  = 1  − i·2-s − 3-s − 4-s + i·5-s + i·6-s + i·8-s + 9-s + 10-s + i·11-s + 12-s − i·15-s + 16-s + 17-s − i·18-s + i·19-s − i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ 0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5971313470 + 0.08841648564i\)
\(L(\frac12)\) \(\approx\) \(0.5971313470 + 0.08841648564i\)
\(L(1)\) \(\approx\) \(0.6936697529 - 0.08586778093i\)
\(L(1)\) \(\approx\) \(0.6936697529 - 0.08586778093i\)

Euler product

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

−30.36235256379769588690836441371, −29.08231883441934029483788916977, −28.01677794982949136426300341850, −27.40200212782066892597448644438, −26.14292774101134690782756064314, −24.76610368572535560130704637034, −24.04842845993512499830329085666, −23.32367840869749348859247473429, −22.05438338091677543734697544025, −21.19967313241630427958440482778, −19.37982069651019617893735395766, −18.158411438232013043640726409646, −17.13092713157679422175474281953, −16.373962799470795582112781293824, −15.61301557309509256390888583366, −13.91437512458049024543337876256, −12.86772167826348890266508981370, −11.75881946912525345280345906824, −10.115640289233802259487868813382, −8.84257983275624655111093410830, −7.607890289957269577787573349911, −6.1096988193255291963287116219, −5.29707701831797870143384365811, −4.091568173825023058291045911171, −0.79098535481316357194219286347, 1.74688801146784934695314227534, 3.487920297345105007216077359806, 4.897970026521705576749466222211, 6.32476991024858219773884900865, 7.81464921538810475779482282019, 9.96142290485052827447587995496, 10.34471011932796142454623721514, 11.74126716824073202979038599380, 12.37223770369539347107106395980, 13.89135232336882052480834930487, 15.08237264085776506170646268615, 16.71709243325565039175562974677, 17.97616187487882684676454836399, 18.45489692930169248311723676399, 19.716559356106776123535234695356, 21.118984524047257926440389980934, 21.99852271909838633126610321644, 22.9707297400933119619317504048, 23.44991048913962279752206623899, 25.37419681944931730115046903627, 26.695057678809649452004603758413, 27.56320634809121601362079656630, 28.458076096948351153722869128646, 29.500670805264426120866868419715, 30.19793506851812591959544143602

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