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Properties

Label 2-91-91.23-c1-0-1
Degree $2$
Conductor $91$
Sign $-0.770 - 0.637i$
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  + 1.37i·2-s + (−1.44 + 2.49i)3-s + 0.0982·4-s + (−0.697 − 0.402i)5-s + (−3.44 − 1.98i)6-s + (0.0699 − 2.64i)7-s + 2.89i·8-s + (−2.65 − 4.59i)9-s + (0.555 − 0.962i)10-s + (4.56 + 2.63i)11-s + (−0.141 + 0.245i)12-s + (−2.36 + 2.72i)13-s + (3.64 + 0.0965i)14-s + (2.01 − 1.16i)15-s − 3.79·16-s + 0.560·17-s + ⋯
L(s)  = 1  + 0.975i·2-s + (−0.831 + 1.44i)3-s + 0.0491·4-s + (−0.312 − 0.180i)5-s + (−1.40 − 0.811i)6-s + (0.0264 − 0.999i)7-s + 1.02i·8-s + (−0.883 − 1.53i)9-s + (0.175 − 0.304i)10-s + (1.37 + 0.794i)11-s + (−0.0408 + 0.0707i)12-s + (−0.656 + 0.754i)13-s + (0.974 + 0.0257i)14-s + (0.519 − 0.299i)15-s − 0.948·16-s + 0.135·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.770 - 0.637i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.283574 + 0.787081i\)
\(L(\frac12)\) \(\approx\) \(0.283574 + 0.787081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.0699 + 2.64i)T \)
13 \( 1 + (2.36 - 2.72i)T \)
good2 \( 1 - 1.37iT - 2T^{2} \)
3 \( 1 + (1.44 - 2.49i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.697 + 0.402i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.56 - 2.63i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.560T + 17T^{2} \)
19 \( 1 + (-5.06 + 2.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.60T + 23T^{2} \)
29 \( 1 + (1.14 + 1.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.01 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.24iT - 37T^{2} \)
41 \( 1 + (-0.803 + 0.463i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.22 + 3.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.32 + 1.92i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.72 + 4.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.9iT - 59T^{2} \)
61 \( 1 + (3.65 + 6.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.36 - 3.67i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.06 + 4.65i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.33 + 2.50i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.68 - 9.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.81iT - 83T^{2} \)
89 \( 1 + 5.00iT - 89T^{2} \)
97 \( 1 + (-9.22 - 5.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.80580041309286978481513320469, −14.02150926619449161545693143832, −11.90147906497088058589852386841, −11.37593922024822406492138277242, −10.07650305779674833762239589117, −9.181671067872071664427453422232, −7.42617068797654676984548774799, −6.45050109268174264704581010809, −4.95940164439146905501957439871, −4.09078694384969952072980629077, 1.36878862363922681752185491903, 3.10132776733547695672129432767, 5.64850931975274013459026071637, 6.67965247218051814049597269846, 7.84091803057780662220014657625, 9.466148012820583884402947854063, 11.04128601822039403791564013436, 11.80248043684492892870384959267, 12.19486738038132871429595125576, 13.14999965882120202699708749666

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