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Properties

Label 2-91-91.25-c1-0-7
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: $\chi_{91} (25, \cdot )$
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

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

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