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Properties

Label 2-91-91.31-c1-0-4
Degree $2$
Conductor $91$
Sign $0.938 - 0.345i$
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  + (1.53 + 0.411i)2-s + (0.436 + 0.251i)3-s + (0.454 + 0.262i)4-s + (−0.0206 + 0.0769i)5-s + (0.565 + 0.565i)6-s + (−2.16 + 1.52i)7-s + (−1.65 − 1.65i)8-s + (−1.37 − 2.37i)9-s + (−0.0632 + 0.109i)10-s + (4.08 − 1.09i)11-s + (0.132 + 0.229i)12-s + (−0.565 + 3.56i)13-s + (−3.94 + 1.44i)14-s + (−0.0283 + 0.0283i)15-s + (−2.38 − 4.13i)16-s + (−2.90 + 5.02i)17-s + ⋯
L(s)  = 1  + (1.08 + 0.290i)2-s + (0.251 + 0.145i)3-s + (0.227 + 0.131i)4-s + (−0.00921 + 0.0343i)5-s + (0.231 + 0.231i)6-s + (−0.818 + 0.575i)7-s + (−0.585 − 0.585i)8-s + (−0.457 − 0.792i)9-s + (−0.0200 + 0.0346i)10-s + (1.23 − 0.329i)11-s + (0.0381 + 0.0661i)12-s + (−0.156 + 0.987i)13-s + (−1.05 + 0.386i)14-s + (−0.00732 + 0.00732i)15-s + (−0.596 − 1.03i)16-s + (−0.704 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.938 - 0.345i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.938 - 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47711 + 0.262955i\)
\(L(\frac12)\) \(\approx\) \(1.47711 + 0.262955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.16 - 1.52i)T \)
13 \( 1 + (0.565 - 3.56i)T \)
good2 \( 1 + (-1.53 - 0.411i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-0.436 - 0.251i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.0206 - 0.0769i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.08 + 1.09i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.90 - 5.02i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.36 + 5.11i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.755 - 0.436i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.362T + 29T^{2} \)
31 \( 1 + (1.34 - 0.361i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.00 + 3.76i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.70 - 7.70i)T + 41iT^{2} \)
43 \( 1 - 2.65iT - 43T^{2} \)
47 \( 1 + (2.79 + 0.748i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.26 - 9.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.573 + 2.14i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.63 + 2.09i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.61 + 9.76i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.65 + 3.65i)T - 71iT^{2} \)
73 \( 1 + (3.08 + 11.4i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.27 + 7.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.91 + 4.91i)T + 83iT^{2} \)
89 \( 1 + (7.78 + 2.08i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.04 + 6.04i)T + 97iT^{2} \)
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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

−14.30507367471248416167131103400, −13.19233317028264346832273735649, −12.30814809325111653262680123301, −11.32728490679005862822166239666, −9.300228050212929447111861193994, −9.057731291859757762088452109425, −6.66146099715790856672286589023, −6.09579800921815562941062462012, −4.36585208538286150272459348784, −3.20750414045285974570165155146, 2.84014819843672381440533003244, 4.15382273003300894328905279123, 5.54785935685510101416598129989, 6.97675222805507916914546798237, 8.454793928931580739065778820088, 9.746036871075969290420347444132, 11.10318782485589158620820871883, 12.23735854195187734567126928508, 13.05041508803505988985559654741, 13.99595839932092202453237930103

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