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Properties

Label 2-91-91.34-c1-0-0
Degree $2$
Conductor $91$
Sign $0.389 - 0.920i$
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  + (−1.45 − 1.45i)2-s + 1.81i·3-s + 2.21i·4-s + (−2.01 + 2.01i)5-s + (2.64 − 2.64i)6-s + (−1.13 + 2.38i)7-s + (0.311 − 0.311i)8-s − 0.311·9-s + 5.84·10-s + (0.451 − 0.451i)11-s − 4.02·12-s + (3.40 − 1.19i)13-s + (5.11 − 1.81i)14-s + (−3.66 − 3.66i)15-s + 3.52·16-s − 4.32·17-s + ⋯
L(s)  = 1  + (−1.02 − 1.02i)2-s + 1.05i·3-s + 1.10i·4-s + (−0.900 + 0.900i)5-s + (1.07 − 1.07i)6-s + (−0.429 + 0.903i)7-s + (0.109 − 0.109i)8-s − 0.103·9-s + 1.84·10-s + (0.136 − 0.136i)11-s − 1.16·12-s + (0.943 − 0.330i)13-s + (1.36 − 0.486i)14-s + (−0.946 − 0.946i)15-s + 0.881·16-s − 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.389 - 0.920i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.389 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382950 + 0.253704i\)
\(L(\frac12)\) \(\approx\) \(0.382950 + 0.253704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.13 - 2.38i)T \)
13 \( 1 + (-3.40 + 1.19i)T \)
good2 \( 1 + (1.45 + 1.45i)T + 2iT^{2} \)
3 \( 1 - 1.81iT - 3T^{2} \)
5 \( 1 + (2.01 - 2.01i)T - 5iT^{2} \)
11 \( 1 + (-0.451 + 0.451i)T - 11iT^{2} \)
17 \( 1 + 4.32T + 17T^{2} \)
19 \( 1 + (3.40 - 3.40i)T - 19iT^{2} \)
23 \( 1 + 0.933iT - 23T^{2} \)
29 \( 1 - 6.33T + 29T^{2} \)
31 \( 1 + (-5.47 + 5.47i)T - 31iT^{2} \)
37 \( 1 + (-2.14 + 2.14i)T - 37iT^{2} \)
41 \( 1 + (-1.81 + 1.81i)T - 41iT^{2} \)
43 \( 1 - 10.4iT - 43T^{2} \)
47 \( 1 + (-5.90 - 5.90i)T + 47iT^{2} \)
53 \( 1 - 3.36T + 53T^{2} \)
59 \( 1 + (0.255 + 0.255i)T + 59iT^{2} \)
61 \( 1 + 7.78iT - 61T^{2} \)
67 \( 1 + (-7.28 - 7.28i)T + 67iT^{2} \)
71 \( 1 + (-5.56 - 5.56i)T + 71iT^{2} \)
73 \( 1 + (8.86 + 8.86i)T + 73iT^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + (-4.30 + 4.30i)T - 83iT^{2} \)
89 \( 1 + (-5.61 - 5.61i)T + 89iT^{2} \)
97 \( 1 + (0.236 - 0.236i)T - 97iT^{2} \)
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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

−14.66842980712798614978897452643, −12.79260158569794243276418745787, −11.59610215321874776925129784036, −10.90773872102485708980811476687, −10.12104077242343463341119190727, −9.057718779604643557384275476127, −8.129723040987991036688666960585, −6.23939874224492284056928273416, −4.04595004440018051472213196042, −2.81146420929664847500361139621, 0.816151103817081694719927913812, 4.27092141553524445919487173591, 6.51077263771438043937067101725, 7.08188394131224620869164390284, 8.242563186318682345480721552789, 8.876086172390961923142255764921, 10.42055631421656241055246977426, 11.92386257804758967180702034176, 12.93622941433560973143171754846, 13.77486074439034149983297155167

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