# 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 $0$

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## 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