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Properties

Label 2-99-33.26-c2-0-4
Degree $2$
Conductor $99$
Sign $0.733 + 0.679i$
Analytic cond. $2.69755$
Root an. cond. $1.64242$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.91 + 2.63i)2-s + (−2.03 − 6.26i)4-s + (−0.936 − 1.28i)5-s + (−3.36 − 10.3i)7-s + (8.01 + 2.60i)8-s + 5.18·10-s + (−5.25 − 9.66i)11-s + (7.47 + 5.43i)13-s + (33.6 + 10.9i)14-s + (−0.868 + 0.630i)16-s + (−2.83 − 3.89i)17-s + (−7.75 + 23.8i)19-s + (−6.17 + 8.49i)20-s + (35.4 + 4.64i)22-s − 36.6i·23-s + ⋯
L(s)  = 1  + (−0.956 + 1.31i)2-s + (−0.509 − 1.56i)4-s + (−0.187 − 0.257i)5-s + (−0.480 − 1.47i)7-s + (1.00 + 0.325i)8-s + 0.518·10-s + (−0.477 − 0.878i)11-s + (0.574 + 0.417i)13-s + (2.40 + 0.781i)14-s + (−0.0542 + 0.0394i)16-s + (−0.166 − 0.229i)17-s + (−0.408 + 1.25i)19-s + (−0.308 + 0.424i)20-s + (1.61 + 0.211i)22-s − 1.59i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.733 + 0.679i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.733 + 0.679i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.452044 - 0.177291i\)
\(L(\frac12)\) \(\approx\) \(0.452044 - 0.177291i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (5.25 + 9.66i)T \)
good2 \( 1 + (1.91 - 2.63i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (0.936 + 1.28i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (3.36 + 10.3i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (-7.47 - 5.43i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (2.83 + 3.89i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (7.75 - 23.8i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + 36.6iT - 529T^{2} \)
29 \( 1 + (23.2 - 7.55i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (47.0 + 34.1i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-3.95 - 12.1i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (11.8 + 3.83i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 19.6T + 1.84e3T^{2} \)
47 \( 1 + (-61.7 - 20.0i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-29.0 + 40.0i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (-22.9 + 7.44i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (35.7 - 26.0i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 33.0T + 4.48e3T^{2} \)
71 \( 1 + (-44.0 - 60.6i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-12.7 - 39.3i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (51.7 + 37.6i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (24.7 + 34.1i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 111. iT - 7.92e3T^{2} \)
97 \( 1 + (-87.4 - 63.5i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−13.87447591748378948053891638001, −12.74383188902313306867126229425, −10.91354465938557519435143801613, −10.09844487046605956645082274417, −8.820446971603992290861127625417, −7.927496287310622857196943797464, −6.89572232460815120021325459310, −5.86875105490812122378900494080, −4.00838268619567529810782452450, −0.49711979763347499265543301967, 2.10662485716598811393729984641, 3.36054935636047164089681570874, 5.55508236791618334306871049585, 7.40333119583578113103352279651, 8.867775128638917046826579174511, 9.372019005345136216510555489560, 10.66263223176414571309939329642, 11.46643845624871546058020518499, 12.50241904584914022726424699077, 13.14124472816624138811463035804

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