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Properties

Label 2-91-91.47-c1-0-2
Degree $2$
Conductor $91$
Sign $-0.0805 - 0.996i$
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  + (0.788 − 0.211i)2-s + (−2.60 + 1.50i)3-s + (−1.15 + 0.666i)4-s + (0.814 + 3.03i)5-s + (−1.73 + 1.73i)6-s + (1.32 − 2.28i)7-s + (−1.92 + 1.92i)8-s + (3.02 − 5.23i)9-s + (1.28 + 2.22i)10-s + (0.491 + 0.131i)11-s + (2.00 − 3.47i)12-s + (1.73 + 3.15i)13-s + (0.562 − 2.08i)14-s + (−6.68 − 6.68i)15-s + (0.221 − 0.382i)16-s + (0.606 + 1.05i)17-s + ⋯
L(s)  = 1  + (0.557 − 0.149i)2-s + (−1.50 + 0.868i)3-s + (−0.577 + 0.333i)4-s + (0.364 + 1.35i)5-s + (−0.709 + 0.709i)6-s + (0.501 − 0.865i)7-s + (−0.680 + 0.680i)8-s + (1.00 − 1.74i)9-s + (0.406 + 0.703i)10-s + (0.148 + 0.0396i)11-s + (0.578 − 1.00i)12-s + (0.481 + 0.876i)13-s + (0.150 − 0.557i)14-s + (−1.72 − 1.72i)15-s + (0.0552 − 0.0957i)16-s + (0.147 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0805 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.0805 - 0.996i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.0805 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.523083 + 0.567049i\)
\(L(\frac12)\) \(\approx\) \(0.523083 + 0.567049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.32 + 2.28i)T \)
13 \( 1 + (-1.73 - 3.15i)T \)
good2 \( 1 + (-0.788 + 0.211i)T + (1.73 - i)T^{2} \)
3 \( 1 + (2.60 - 1.50i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.814 - 3.03i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.491 - 0.131i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.606 - 1.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.461 + 1.72i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.51 - 2.60i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.64T + 29T^{2} \)
31 \( 1 + (3.64 + 0.976i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.715 - 2.66i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.55 + 5.55i)T - 41iT^{2} \)
43 \( 1 + 7.46iT - 43T^{2} \)
47 \( 1 + (4.73 - 1.26i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.30 - 7.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.648 - 2.41i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.09 + 5.25i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.91 - 7.15i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.840 - 0.840i)T + 71iT^{2} \)
73 \( 1 + (-0.632 + 2.36i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.20 + 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.31 + 7.31i)T - 83iT^{2} \)
89 \( 1 + (-9.42 + 2.52i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.93 + 2.93i)T - 97iT^{2} \)
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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

−14.34235672255093670145373489656, −13.42012738224398630005180420472, −11.95300212864443964968896453019, −11.09293097230322255009060303490, −10.50879501487767742034521581012, −9.248514455543318984949869546643, −7.14178427236316173337335855452, −5.99915607752218747404191866892, −4.71414913519175166630817140955, −3.65626233030551999834157560968, 1.09825545910544844586515134276, 4.83175250236778794339499871371, 5.43350946245622596981700166609, 6.27889137393267868627866663020, 8.176704528398513492389127908437, 9.357584180896763508634771570796, 10.90143930460902584058486132986, 12.13677850468915770990630404535, 12.74146228508745632848403673758, 13.32078915330641954488920190741

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