Properties

Label 2-91-91.6-c1-0-5
Degree $2$
Conductor $91$
Sign $0.0899 + 0.995i$
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  + (−0.607 − 2.26i)2-s + (2.39 + 1.38i)3-s + (−3.03 + 1.75i)4-s + (−1.12 − 1.12i)5-s + (1.67 − 6.25i)6-s + (1.68 − 2.04i)7-s + (2.50 + 2.50i)8-s + (2.30 + 4.00i)9-s + (−1.87 + 3.23i)10-s + (−3.03 + 0.813i)11-s − 9.68·12-s + (1.04 + 3.44i)13-s + (−5.64 − 2.57i)14-s + (−1.13 − 4.24i)15-s + (0.649 − 1.12i)16-s + (−0.320 − 0.555i)17-s + ⋯
L(s)  = 1  + (−0.429 − 1.60i)2-s + (1.38 + 0.796i)3-s + (−1.51 + 0.877i)4-s + (−0.503 − 0.503i)5-s + (0.684 − 2.55i)6-s + (0.636 − 0.771i)7-s + (0.886 + 0.886i)8-s + (0.769 + 1.33i)9-s + (−0.591 + 1.02i)10-s + (−0.914 + 0.245i)11-s − 2.79·12-s + (0.290 + 0.956i)13-s + (−1.51 − 0.689i)14-s + (−0.293 − 1.09i)15-s + (0.162 − 0.281i)16-s + (−0.0778 − 0.134i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0899 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0899 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.0899 + 0.995i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.0899 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.772838 - 0.706177i\)
\(L(\frac12)\) \(\approx\) \(0.772838 - 0.706177i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.68 + 2.04i)T \)
13 \( 1 + (-1.04 - 3.44i)T \)
good2 \( 1 + (0.607 + 2.26i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (-2.39 - 1.38i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.12 + 1.12i)T + 5iT^{2} \)
11 \( 1 + (3.03 - 0.813i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.320 + 0.555i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.04 - 7.61i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.126 + 0.0730i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.49 + 2.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.73 + 4.73i)T + 31iT^{2} \)
37 \( 1 + (-3.75 + 1.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.60 - 1.50i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.42 + 1.40i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.22 + 2.22i)T - 47iT^{2} \)
53 \( 1 + 7.32T + 53T^{2} \)
59 \( 1 + (-4.00 - 1.07i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.90 + 2.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.366 + 1.36i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-13.8 - 3.70i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.99 - 4.99i)T - 73iT^{2} \)
79 \( 1 + 0.632T + 79T^{2} \)
83 \( 1 + (1.07 + 1.07i)T + 83iT^{2} \)
89 \( 1 + (3.51 + 13.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.0487 - 0.181i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.72203178484628949928200402373, −12.69422884548310813127107933794, −11.47981159509533553611504578630, −10.41048914587570372186955371544, −9.698866607901905900231682865439, −8.529028026603728345675561625099, −7.903906498128033190006426259378, −4.42682148311242025195000752781, −3.75579747214940077069931917991, −2.07880069085708507344483111632, 2.84686576502905856294429944377, 5.24466427734244306288666517062, 6.80327381428656092582227142120, 7.72956943581296285150304973726, 8.372004808116336534429087588558, 9.143129373043839850650793355731, 10.98530987473191255290443712363, 12.78045091996655695139348929269, 13.68059270282065351840534395734, 14.72514891250634866789775262238

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