Properties

Label 2-91-91.51-c1-0-2
Degree $2$
Conductor $91$
Sign $0.644 - 0.764i$
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  + (0.929 + 0.536i)2-s + (1.21 + 2.10i)3-s + (−0.424 − 0.734i)4-s + (−0.541 − 0.312i)5-s + 2.60i·6-s + (−2.34 − 1.21i)7-s − 3.05i·8-s + (−1.45 + 2.52i)9-s + (−0.335 − 0.581i)10-s + (0.613 − 0.354i)11-s + (1.03 − 1.78i)12-s + (0.848 + 3.50i)13-s + (−1.53 − 2.39i)14-s − 1.52i·15-s + (0.791 − 1.37i)16-s + (−1.67 − 2.89i)17-s + ⋯
L(s)  = 1  + (0.657 + 0.379i)2-s + (0.701 + 1.21i)3-s + (−0.212 − 0.367i)4-s + (−0.242 − 0.139i)5-s + 1.06i·6-s + (−0.888 − 0.459i)7-s − 1.08i·8-s + (−0.485 + 0.840i)9-s + (−0.106 − 0.183i)10-s + (0.185 − 0.106i)11-s + (0.297 − 0.515i)12-s + (0.235 + 0.971i)13-s + (−0.409 − 0.638i)14-s − 0.392i·15-s + (0.197 − 0.342i)16-s + (−0.405 − 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.644 - 0.764i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.644 - 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24424 + 0.578762i\)
\(L(\frac12)\) \(\approx\) \(1.24424 + 0.578762i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.34 + 1.21i)T \)
13 \( 1 + (-0.848 - 3.50i)T \)
good2 \( 1 + (-0.929 - 0.536i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.21 - 2.10i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.541 + 0.312i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.613 + 0.354i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.67 + 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.50 - 2.60i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.21 - 3.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.59T + 29T^{2} \)
31 \( 1 + (3.80 - 2.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.366 - 0.211i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.01iT - 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (-6.99 - 4.03i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.348 - 0.603i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.54 - 4.93i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.34 - 4.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.02 + 5.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 + (-4.40 + 2.54i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.95 + 3.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + (11.5 + 6.68i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.202iT - 97T^{2} \)
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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

−13.98380038323636205769125263887, −13.89695254795108868760744289448, −12.33255416941330628578375520780, −10.77620534679313418174839304994, −9.550559155594965595214728577865, −9.239299215335371171907428187080, −7.29459852351745014542839532165, −5.82421623975259621374630137220, −4.33113533090819161501074838426, −3.58507607506503269670956576229, 2.49551920172639659712688860027, 3.66260551446104778064162822819, 5.73884135227777525111381772771, 7.24157543851079758923519369850, 8.200599192047528032156570646210, 9.303289440348577440331990821192, 11.12793614436320300979338566134, 12.36240209561281607896235722014, 12.90637015867166192201219831698, 13.55699793554073994252776714857

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