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Properties

Label 1-91-91.34-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: $\chi_{91} (34, \cdot )$
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

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

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