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Properties

Label 2-91-91.80-c1-0-3
Degree $2$
Conductor $91$
Sign $-0.225 + 0.974i$
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  + (−0.521 − 1.94i)2-s + 1.44i·3-s + (−1.78 + 1.03i)4-s + (0.849 − 3.16i)5-s + (2.81 − 0.753i)6-s + (−1.41 − 2.23i)7-s + (0.0872 + 0.0872i)8-s + 0.915·9-s − 6.61·10-s + (4.20 + 4.20i)11-s + (−1.48 − 2.57i)12-s + (−2.81 + 2.25i)13-s + (−3.60 + 3.92i)14-s + (4.57 + 1.22i)15-s + (−1.93 + 3.35i)16-s + (−0.314 − 0.544i)17-s + ⋯
L(s)  = 1  + (−0.368 − 1.37i)2-s + 0.833i·3-s + (−0.892 + 0.515i)4-s + (0.379 − 1.41i)5-s + (1.14 − 0.307i)6-s + (−0.536 − 0.844i)7-s + (0.0308 + 0.0308i)8-s + 0.305·9-s − 2.09·10-s + (1.26 + 1.26i)11-s + (−0.429 − 0.743i)12-s + (−0.779 + 0.626i)13-s + (−0.964 + 1.04i)14-s + (1.18 + 0.316i)15-s + (−0.484 + 0.838i)16-s + (−0.0762 − 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.225 + 0.974i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.225 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531144 - 0.667849i\)
\(L(\frac12)\) \(\approx\) \(0.531144 - 0.667849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.41 + 2.23i)T \)
13 \( 1 + (2.81 - 2.25i)T \)
good2 \( 1 + (0.521 + 1.94i)T + (-1.73 + i)T^{2} \)
3 \( 1 - 1.44iT - 3T^{2} \)
5 \( 1 + (-0.849 + 3.16i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.20 - 4.20i)T + 11iT^{2} \)
17 \( 1 + (0.314 + 0.544i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.521 - 0.521i)T + 19iT^{2} \)
23 \( 1 + (-3.93 - 2.27i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.33 + 2.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.06 - 0.285i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.44 - 0.655i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.746 + 2.78i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.49 - 2.01i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.10 + 1.90i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.89 + 6.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.919 - 0.246i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 1.08iT - 61T^{2} \)
67 \( 1 + (3.81 - 3.81i)T - 67iT^{2} \)
71 \( 1 + (-0.590 - 2.20i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.59 + 5.94i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.08 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.59 + 3.59i)T + 83iT^{2} \)
89 \( 1 + (-1.05 - 3.94i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.41 - 2.25i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.29334934727824438500812578289, −12.54208367024162806854662454310, −11.67172769731436606042052276751, −10.25546309484988266077008691563, −9.507723589585804408710391002822, −9.166826613928162805993567290034, −6.98802990914662599060322674680, −4.72412131454047664994483401584, −3.91556667322079176693901165232, −1.52365885279982361924781106131, 2.88183043815839074187097469123, 5.78706148763158948371301045294, 6.57517623142664197083647757687, 7.21803128583762168224160113623, 8.565603739321142032149698991320, 9.691593454606216990019226813485, 11.19503476932553917845801877596, 12.43160329244087854902284736881, 13.75329963044911748780463372984, 14.59846313402771622194642472236

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