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Properties

Label 2-91-91.81-c1-0-3
Degree $2$
Conductor $91$
Sign $0.835 + 0.549i$
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  + (0.425 − 0.737i)2-s + 0.661·3-s + (0.637 + 1.10i)4-s + (−1.72 − 2.98i)5-s + (0.281 − 0.487i)6-s + (1.82 + 1.91i)7-s + 2.78·8-s − 2.56·9-s − 2.92·10-s − 0.897·11-s + (0.421 + 0.730i)12-s + (−3.07 + 1.88i)13-s + (2.18 − 0.525i)14-s + (−1.13 − 1.97i)15-s + (−0.0891 + 0.154i)16-s + (−0.968 − 1.67i)17-s + ⋯
L(s)  = 1  + (0.300 − 0.521i)2-s + 0.381·3-s + (0.318 + 0.552i)4-s + (−0.769 − 1.33i)5-s + (0.114 − 0.198i)6-s + (0.688 + 0.725i)7-s + 0.985·8-s − 0.854·9-s − 0.926·10-s − 0.270·11-s + (0.121 + 0.210i)12-s + (−0.852 + 0.522i)13-s + (0.585 − 0.140i)14-s + (−0.293 − 0.508i)15-s + (−0.0222 + 0.0386i)16-s + (−0.234 − 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17060 - 0.350081i\)
\(L(\frac12)\) \(\approx\) \(1.17060 - 0.350081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.82 - 1.91i)T \)
13 \( 1 + (3.07 - 1.88i)T \)
good2 \( 1 + (-0.425 + 0.737i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 0.661T + 3T^{2} \)
5 \( 1 + (1.72 + 2.98i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.897T + 11T^{2} \)
17 \( 1 + (0.968 + 1.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 1.03T + 19T^{2} \)
23 \( 1 + (2.82 - 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.917 - 1.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.56 + 7.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.30 + 9.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.66 - 4.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.95 + 3.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.59 + 6.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.69 + 8.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.255 - 0.442i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.43T + 61T^{2} \)
67 \( 1 + 8.44T + 67T^{2} \)
71 \( 1 + (-1.72 + 2.98i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.45 - 9.44i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.04 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.51T + 83T^{2} \)
89 \( 1 + (6.80 - 11.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.253 - 0.438i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.78353178314597347441636827529, −12.71433126338250883585854738774, −11.76456721465226781299846671936, −11.45232025200309658110770882044, −9.387815397312900323903601476874, −8.328433463263302207206998795737, −7.64071654156067761968538923123, −5.31339248127785475489640391337, −4.13412758953910039249696777954, −2.37414593614358078020455315650, 2.80729778271699423842019959130, 4.59994424967326469857119611745, 6.24132413699861190837959589029, 7.37622698324790937842623975584, 8.128445280838652079945059846146, 10.23783547267479541796982520916, 10.83610598699277088580969288796, 11.85223180675114560978710229113, 13.69426330326917635776142310814, 14.52875663996914739134406325524

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