Properties

Label 2-91-91.54-c1-0-2
Degree $2$
Conductor $91$
Sign $0.0411 - 0.999i$
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  + (−1.42 + 1.42i)2-s + (1.25 + 0.721i)3-s − 2.06i·4-s + (3.16 − 0.849i)5-s + (−2.81 + 0.753i)6-s + (−0.111 + 2.64i)7-s + (0.0872 + 0.0872i)8-s + (−0.457 − 0.793i)9-s + (−3.30 + 5.72i)10-s + (−5.74 + 1.53i)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 + 3.87·16-s − 0.628·17-s + ⋯
L(s)  = 1  + (−1.00 + 1.00i)2-s + (0.721 + 0.416i)3-s − 1.03i·4-s + (1.41 − 0.379i)5-s + (−1.14 + 0.307i)6-s + (−0.0421 + 0.999i)7-s + (0.0308 + 0.0308i)8-s + (−0.152 − 0.264i)9-s + (−1.04 + 1.81i)10-s + (−1.73 + 0.463i)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 + 0.968·16-s − 0.152·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0411 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0411 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.0411 - 0.999i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.0411 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.601921 + 0.577626i\)
\(L(\frac12)\) \(\approx\) \(0.601921 + 0.577626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.111 - 2.64i)T \)
13 \( 1 + (-2.81 + 2.25i)T \)
good2 \( 1 + (1.42 - 1.42i)T - 2iT^{2} \)
3 \( 1 + (-1.25 - 0.721i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.16 + 0.849i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (5.74 - 1.53i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 0.628T + 17T^{2} \)
19 \( 1 + (0.191 - 0.712i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 4.54iT - 23T^{2} \)
29 \( 1 + (1.33 + 2.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.285 - 1.06i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.79 - 1.79i)T + 37iT^{2} \)
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 \( 1 + (1.90 + 7.10i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.89 - 6.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.673 + 0.673i)T - 59iT^{2} \)
61 \( 1 + (0.943 - 0.544i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.39 + 5.21i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.590 - 2.20i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (5.94 + 1.59i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.08 - 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.59 - 3.59i)T + 83iT^{2} \)
89 \( 1 + (2.88 - 2.88i)T - 89iT^{2} \)
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

−14.80051200085141983666658530904, −13.45301076112332470216686834473, −12.55325899098142292771252282247, −10.39194482845576537803820426439, −9.623427539153283328734730567076, −8.763353389606734682430584552827, −8.050082379385469964947032260323, −6.24716863177539271612242046079, −5.39803121277205079689560063600, −2.63506599991673285221422834252, 1.83410754671187552947916835271, 3.00585260621055337134555787911, 5.67521263148767595979259355703, 7.42129238259283802058873636811, 8.528311893859910551479201769517, 9.605890451210837088978532108451, 10.55914964983302306544133087756, 11.10414016409495126351615936127, 13.11818395866057672586430332385, 13.52599844218735111783735557383

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