Properties

Label 2-99-99.97-c1-0-5
Degree $2$
Conductor $99$
Sign $0.618 - 0.785i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
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  + (1.55 + 1.73i)2-s + (0.0128 − 1.73i)3-s + (−0.357 + 3.40i)4-s + (−0.212 + 0.235i)5-s + (3.01 − 2.67i)6-s + (−1.29 + 0.576i)7-s + (−2.68 + 1.94i)8-s + (−2.99 − 0.0443i)9-s − 0.738·10-s + (2.21 − 2.46i)11-s + (5.89 + 0.663i)12-s + (−4.10 + 0.872i)13-s + (−3.01 − 1.34i)14-s + (0.405 + 0.370i)15-s + (−0.857 − 0.182i)16-s + (−1.92 − 5.93i)17-s + ⋯
L(s)  = 1  + (1.10 + 1.22i)2-s + (0.00739 − 0.999i)3-s + (−0.178 + 1.70i)4-s + (−0.0948 + 0.105i)5-s + (1.23 − 1.09i)6-s + (−0.489 + 0.218i)7-s + (−0.948 + 0.689i)8-s + (−0.999 − 0.0147i)9-s − 0.233·10-s + (0.669 − 0.743i)11-s + (1.70 + 0.191i)12-s + (−1.13 + 0.242i)13-s + (−0.806 − 0.359i)14-s + (0.104 + 0.0956i)15-s + (−0.214 − 0.0455i)16-s + (−0.467 − 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.618 - 0.785i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.618 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38695 + 0.673095i\)
\(L(\frac12)\) \(\approx\) \(1.38695 + 0.673095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0128 + 1.73i)T \)
11 \( 1 + (-2.21 + 2.46i)T \)
good2 \( 1 + (-1.55 - 1.73i)T + (-0.209 + 1.98i)T^{2} \)
5 \( 1 + (0.212 - 0.235i)T + (-0.522 - 4.97i)T^{2} \)
7 \( 1 + (1.29 - 0.576i)T + (4.68 - 5.20i)T^{2} \)
13 \( 1 + (4.10 - 0.872i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (1.92 + 5.93i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.50 - 1.82i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.26 - 3.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.67 + 1.19i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (-9.33 + 1.98i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (0.685 + 0.498i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.53 + 1.57i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (1.77 - 3.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.778 - 7.41i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (-2.06 + 6.35i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.26 + 12.0i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (4.51 + 0.960i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (4.98 + 8.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.905 - 2.78i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.43 + 2.49i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.26 + 2.51i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (-13.1 - 2.80i)T + (75.8 + 33.7i)T^{2} \)
89 \( 1 - 2.69T + 89T^{2} \)
97 \( 1 + (3.51 + 3.90i)T + (-10.1 + 96.4i)T^{2} \)
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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

−13.98341164429027116199073202218, −13.35198211078270460584556430219, −12.30915520556912141910764458026, −11.49759322519798933658422108210, −9.338022422405901516008855343504, −7.973167317336159296019446292582, −6.95682730035359040057088851158, −6.23062284905689317348130424093, −4.93462089264392114748658455925, −3.08968466480678245033060669341, 2.63286753344180303770175180609, 4.10301904222029747515595786205, 4.81650553380669411505497329891, 6.44003757991444470468090351091, 8.664958045701448653150672186889, 10.14315125982924979266763429288, 10.42152939710298836918206071587, 11.85271345910457528813058423086, 12.48989294175452895591200333884, 13.62651308107051943991500315481

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