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Properties

Label 2-91-91.19-c1-0-5
Degree $2$
Conductor $91$
Sign $0.986 - 0.164i$
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  + (2.14 + 0.573i)2-s − 1.10i·3-s + (2.52 + 1.45i)4-s + (−2.92 + 0.784i)5-s + (0.635 − 2.37i)6-s + (−2.64 + 0.0746i)7-s + (1.43 + 1.43i)8-s + 1.77·9-s − 6.72·10-s + (−0.516 − 0.516i)11-s + (1.61 − 2.79i)12-s + (2.65 + 2.44i)13-s + (−5.70 − 1.35i)14-s + (0.869 + 3.24i)15-s + (−0.668 − 1.15i)16-s + (1.62 − 2.81i)17-s + ⋯
L(s)  = 1  + (1.51 + 0.405i)2-s − 0.639i·3-s + (1.26 + 0.728i)4-s + (−1.31 + 0.351i)5-s + (0.259 − 0.968i)6-s + (−0.999 + 0.0282i)7-s + (0.506 + 0.506i)8-s + 0.590·9-s − 2.12·10-s + (−0.155 − 0.155i)11-s + (0.466 − 0.807i)12-s + (0.735 + 0.678i)13-s + (−1.52 − 0.362i)14-s + (0.224 + 0.838i)15-s + (−0.167 − 0.289i)16-s + (0.393 − 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.986 - 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65039 + 0.136415i\)
\(L(\frac12)\) \(\approx\) \(1.65039 + 0.136415i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.64 - 0.0746i)T \)
13 \( 1 + (-2.65 - 2.44i)T \)
good2 \( 1 + (-2.14 - 0.573i)T + (1.73 + i)T^{2} \)
3 \( 1 + 1.10iT - 3T^{2} \)
5 \( 1 + (2.92 - 0.784i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.516 + 0.516i)T + 11iT^{2} \)
17 \( 1 + (-1.62 + 2.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.82 - 2.82i)T + 19iT^{2} \)
23 \( 1 + (5.16 - 2.98i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.78 + 4.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.07 + 3.99i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.55 - 9.52i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.46 - 0.660i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.73 - 3.30i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.454 + 1.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.37 - 5.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.853 - 3.18i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 7.27iT - 61T^{2} \)
67 \( 1 + (-4.59 + 4.59i)T - 67iT^{2} \)
71 \( 1 + (8.06 + 2.16i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.08 - 1.62i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.87 + 13.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.4 + 10.4i)T + 83iT^{2} \)
89 \( 1 + (9.05 + 2.42i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.37 - 8.86i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.79609786670525505736254407664, −13.33375343481183742261805906215, −11.94196632359978049571894282101, −11.84497309505664569021586541866, −9.891659272492360849383818043872, −7.901620606135390873153433055144, −6.98682686017887803095349327199, −6.05791661785844315016872633468, −4.24503883468051851754398925922, −3.27246128692359169090624983409, 3.38669615135332944653934283095, 4.06137507454810546660755008084, 5.33393793333796954163156122947, 6.87313775286664358027741708494, 8.482174599190644322056693778297, 10.12760428116545717446403618106, 11.09763329055958280589149951678, 12.39930025204061200067045148940, 12.70119885643165145896421849946, 13.94986841537874728031092074672

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