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Properties

Label 2-91-91.9-c1-0-1
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: $\chi_{91} (9, \cdot )$
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

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

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