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Properties

Label 2-91-91.51-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.451 + 0.892i$
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  + (−0.287 − 0.166i)2-s + (−0.729 − 1.26i)3-s + (−0.944 − 1.63i)4-s + (−1.25 − 0.722i)5-s + 0.485i·6-s + (−2.26 + 1.36i)7-s + 1.29i·8-s + (0.434 − 0.752i)9-s + (0.240 + 0.416i)10-s + (5.15 − 2.97i)11-s + (−1.37 + 2.38i)12-s + (1.88 − 3.07i)13-s + (0.879 − 0.0178i)14-s + 2.11i·15-s + (−1.67 + 2.90i)16-s + (2.16 + 3.74i)17-s + ⋯
L(s)  = 1  + (−0.203 − 0.117i)2-s + (−0.421 − 0.729i)3-s + (−0.472 − 0.818i)4-s + (−0.559 − 0.323i)5-s + 0.198i·6-s + (−0.855 + 0.517i)7-s + 0.457i·8-s + (0.144 − 0.250i)9-s + (0.0759 + 0.131i)10-s + (1.55 − 0.897i)11-s + (−0.398 + 0.689i)12-s + (0.524 − 0.851i)13-s + (0.234 − 0.00477i)14-s + 0.544i·15-s + (−0.418 + 0.725i)16-s + (0.524 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327949 - 0.533162i\)
\(L(\frac12)\) \(\approx\) \(0.327949 - 0.533162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.26 - 1.36i)T \)
13 \( 1 + (-1.88 + 3.07i)T \)
good2 \( 1 + (0.287 + 0.166i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.729 + 1.26i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.25 + 0.722i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.15 + 2.97i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.16 - 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.69 + 0.978i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.270 - 0.467i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.15T + 29T^{2} \)
31 \( 1 + (5.28 - 3.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.95 - 4.01i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.55iT - 41T^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 + (-5.42 - 3.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.38 - 2.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.737 + 0.425i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 - 5.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.854 - 0.493i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.76iT - 71T^{2} \)
73 \( 1 + (-7.91 + 4.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.0655 + 0.113i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.66iT - 83T^{2} \)
89 \( 1 + (-8.41 - 4.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.58iT - 97T^{2} \)
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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

−13.61246559949443063989164083446, −12.56193058491671707736548168657, −11.79598137203079043214311127976, −10.52339582705287532156831761802, −9.244078615350286252565623463698, −8.364125922707267870142610900369, −6.48628070345805857111518212720, −5.84093105180605494087032141823, −3.82651212048661204288555503922, −0.964227888640468369760857581907, 3.70677513696973079012478685328, 4.43059854632253306029957502812, 6.62571673564314392268318103051, 7.57334326538579638256809549523, 9.209984742982313074951022900272, 9.863314128251826168816946781503, 11.31069651555224579824571088784, 12.16419820404556397572983747519, 13.36073715427898994230476379753, 14.46272136008758186971020841768

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