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Properties

Label 2-91-7.6-c2-0-15
Degree $2$
Conductor $91$
Sign $0.635 + 0.772i$
Analytic cond. $2.47957$
Root an. cond. $1.57466$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.18·2-s − 5.40i·3-s + 6.13·4-s + 5.26i·5-s − 17.2i·6-s + (−5.40 + 4.44i)7-s + 6.81·8-s − 20.1·9-s + 16.7i·10-s + 18.3·11-s − 33.1i·12-s + 3.60i·13-s + (−17.2 + 14.1i)14-s + 28.4·15-s − 2.86·16-s − 15.4i·17-s + ⋯
L(s)  = 1  + 1.59·2-s − 1.80i·3-s + 1.53·4-s + 1.05i·5-s − 2.86i·6-s + (−0.772 + 0.635i)7-s + 0.851·8-s − 2.24·9-s + 1.67i·10-s + 1.66·11-s − 2.76i·12-s + 0.277i·13-s + (−1.22 + 1.01i)14-s + 1.89·15-s − 0.179·16-s − 0.908i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.635 + 0.772i$
Analytic conductor: \(2.47957\)
Root analytic conductor: \(1.57466\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1),\ 0.635 + 0.772i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.41563 - 1.14093i\)
\(L(\frac12)\) \(\approx\) \(2.41563 - 1.14093i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (5.40 - 4.44i)T \)
13 \( 1 - 3.60iT \)
good2 \( 1 - 3.18T + 4T^{2} \)
3 \( 1 + 5.40iT - 9T^{2} \)
5 \( 1 - 5.26iT - 25T^{2} \)
11 \( 1 - 18.3T + 121T^{2} \)
17 \( 1 + 15.4iT - 289T^{2} \)
19 \( 1 - 6.11iT - 361T^{2} \)
23 \( 1 + 8.74T + 529T^{2} \)
29 \( 1 + 29.7T + 841T^{2} \)
31 \( 1 + 8.83iT - 961T^{2} \)
37 \( 1 + 18.0T + 1.36e3T^{2} \)
41 \( 1 + 38.4iT - 1.68e3T^{2} \)
43 \( 1 - 9.39T + 1.84e3T^{2} \)
47 \( 1 - 43.0iT - 2.20e3T^{2} \)
53 \( 1 - 33.0T + 2.80e3T^{2} \)
59 \( 1 - 26.8iT - 3.48e3T^{2} \)
61 \( 1 - 6.66iT - 3.72e3T^{2} \)
67 \( 1 - 17.7T + 4.48e3T^{2} \)
71 \( 1 - 95.2T + 5.04e3T^{2} \)
73 \( 1 + 29.7iT - 5.32e3T^{2} \)
79 \( 1 - 59.0T + 6.24e3T^{2} \)
83 \( 1 + 110. iT - 6.88e3T^{2} \)
89 \( 1 - 147. iT - 7.92e3T^{2} \)
97 \( 1 + 60.6iT - 9.40e3T^{2} \)
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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

−13.79056840998419630740078902020, −12.66740468153561913672216920960, −11.98730635600602298433395469978, −11.33043911898006896043360153475, −9.122298031029243124716680953016, −7.20462864611570754376060273213, −6.58123632197727873531185609190, −5.83551433704158287825689619270, −3.48669640044135400217359993886, −2.26153975652716231368810558912, 3.59965308642451937708153695442, 4.13633066073782760613440957509, 5.23020681434549759610003647746, 6.40589643072184774691979994490, 8.817319615617526635639328039426, 9.691375168340726023965131459742, 10.95510261445196726994057655928, 12.02460595907068410302179366732, 13.04584364769251715071074484502, 14.14676209236120723052520481037

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