Properties

Label 2-91-91.59-c1-0-0
Degree $2$
Conductor $91$
Sign $0.836 - 0.547i$
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  + (−1.72 − 1.72i)2-s + (−1.70 + 0.985i)3-s + 3.92i·4-s + (0.227 + 0.0608i)5-s + (4.63 + 1.24i)6-s + (1.27 + 2.31i)7-s + (3.30 − 3.30i)8-s + (0.442 − 0.766i)9-s + (−0.286 − 0.495i)10-s + (2.76 + 0.740i)11-s + (−3.86 − 6.69i)12-s + (−2.03 + 2.97i)13-s + (1.79 − 6.18i)14-s + (−0.447 + 0.119i)15-s − 3.53·16-s − 4.03·17-s + ⋯
L(s)  = 1  + (−1.21 − 1.21i)2-s + (−0.985 + 0.568i)3-s + 1.96i·4-s + (0.101 + 0.0272i)5-s + (1.89 + 0.506i)6-s + (0.481 + 0.876i)7-s + (1.16 − 1.16i)8-s + (0.147 − 0.255i)9-s + (−0.0904 − 0.156i)10-s + (0.833 + 0.223i)11-s + (−1.11 − 1.93i)12-s + (−0.563 + 0.826i)13-s + (0.480 − 1.65i)14-s + (−0.115 + 0.0309i)15-s − 0.882·16-s − 0.977·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.836 - 0.547i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.836 - 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360350 + 0.107452i\)
\(L(\frac12)\) \(\approx\) \(0.360350 + 0.107452i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.27 - 2.31i)T \)
13 \( 1 + (2.03 - 2.97i)T \)
good2 \( 1 + (1.72 + 1.72i)T + 2iT^{2} \)
3 \( 1 + (1.70 - 0.985i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.227 - 0.0608i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.76 - 0.740i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 4.03T + 17T^{2} \)
19 \( 1 + (-1.42 - 5.30i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 6.05iT - 23T^{2} \)
29 \( 1 + (-3.54 + 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.595 - 2.22i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-7.34 + 7.34i)T - 37iT^{2} \)
41 \( 1 + (0.0713 + 0.266i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.91 + 2.25i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.842 - 3.14i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.96 - 6.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.514 + 0.514i)T + 59iT^{2} \)
61 \( 1 + (-1.97 - 1.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.688 + 2.57i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.362 - 1.35i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-13.0 + 3.50i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.91 + 3.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.19 + 4.19i)T - 83iT^{2} \)
89 \( 1 + (-2.44 - 2.44i)T + 89iT^{2} \)
97 \( 1 + (-2.61 - 0.701i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.10411653068744538668113551760, −12.28482883921731302811688776165, −11.71826447245725925371591530919, −11.07484372741069485297765233111, −9.852306893221898791851342295306, −9.194732154284113946392017200410, −7.86727383242619300572777260923, −5.96234815167346242772263030080, −4.26968469625605425167642262885, −2.07090760794064391748926953315, 0.78833175204379473140735539350, 4.98337545971446518303935498451, 6.39934366158846451560198160682, 7.01062520053824977929776872288, 8.181390557251988753400356424975, 9.400206096345534907622019295858, 10.63783978898472541581038859233, 11.52667541946837131279775239282, 13.01980813747962515749924033078, 14.33122606386237934966196879479

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