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Properties

Label 2-91-91.25-c1-0-0
Degree $2$
Conductor $91$
Sign $-0.695 + 0.718i$
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  + (−1.97 + 1.14i)2-s + (−1.57 + 2.72i)3-s + (1.61 − 2.78i)4-s + (−1.84 + 1.06i)5-s − 7.19i·6-s + (2.62 + 0.331i)7-s + 2.78i·8-s + (−3.46 − 5.99i)9-s + (2.42 − 4.20i)10-s + (−0.267 − 0.154i)11-s + (5.07 + 8.78i)12-s + (−3.22 + 1.62i)13-s + (−5.57 + 2.34i)14-s − 6.69i·15-s + (0.0349 + 0.0605i)16-s + (−0.887 + 1.53i)17-s + ⋯
L(s)  = 1  + (−1.39 + 0.807i)2-s + (−0.909 + 1.57i)3-s + (0.805 − 1.39i)4-s + (−0.823 + 0.475i)5-s − 2.93i·6-s + (0.992 + 0.125i)7-s + 0.985i·8-s + (−1.15 − 1.99i)9-s + (0.767 − 1.32i)10-s + (−0.0805 − 0.0465i)11-s + (1.46 + 2.53i)12-s + (−0.893 + 0.449i)13-s + (−1.48 + 0.626i)14-s − 1.72i·15-s + (0.00874 + 0.0151i)16-s + (−0.215 + 0.372i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.695 + 0.718i$
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: \((2,\ 91,\ (\ :1/2),\ -0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0935059 - 0.220858i\)
\(L(\frac12)\) \(\approx\) \(0.0935059 - 0.220858i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.62 - 0.331i)T \)
13 \( 1 + (3.22 - 1.62i)T \)
good2 \( 1 + (1.97 - 1.14i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.57 - 2.72i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.84 - 1.06i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.267 + 0.154i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.887 - 1.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.54 - 0.890i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.575 - 0.996i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.01T + 29T^{2} \)
31 \( 1 + (3.98 + 2.30i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.79 - 2.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.72iT - 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 + (-8.24 + 4.75i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.72 - 6.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.03 - 4.06i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.72 - 2.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.9 - 6.30i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.35iT - 71T^{2} \)
73 \( 1 + (10.2 + 5.94i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.96 - 6.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.2iT - 83T^{2} \)
89 \( 1 + (1.43 - 0.829i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.66iT - 97T^{2} \)
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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

−15.14338861842581494661291855189, −14.65390829121163923988435903206, −11.92261394369353858914981226122, −11.08558031731229090608118847061, −10.35026104963766537936659286930, −9.326533230173542393602306970810, −8.242690799041873408724823606066, −6.98108708132668289216333721760, −5.51297583246194646429381785208, −4.15645416054321842892882926793, 0.50021622570622079972071439271, 2.09050588412364098663800617053, 5.13240259690263190437673408152, 7.14448470408302776255186859144, 7.83578515051761164202203797217, 8.688265658914622979385461800110, 10.52800103623174862627226967363, 11.37169116678873163179038898205, 12.07620807063449957660975116230, 12.71882779691763632283957339050

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