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Properties

Label 2-99-9.7-c1-0-2
Degree $2$
Conductor $99$
Sign $-0.357 - 0.934i$
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  + (−1.23 + 2.13i)2-s + (1.66 − 0.461i)3-s + (−2.04 − 3.54i)4-s + (1.21 + 2.10i)5-s + (−1.07 + 4.14i)6-s + (−1.16 + 2.02i)7-s + 5.17·8-s + (2.57 − 1.54i)9-s − 6.01·10-s + (−0.5 + 0.866i)11-s + (−5.05 − 4.97i)12-s + (−2.35 − 4.07i)13-s + (−2.88 − 5.00i)14-s + (3.00 + 2.95i)15-s + (−2.29 + 3.97i)16-s − 3.20·17-s + ⋯
L(s)  = 1  + (−0.873 + 1.51i)2-s + (0.963 − 0.266i)3-s + (−1.02 − 1.77i)4-s + (0.544 + 0.943i)5-s + (−0.438 + 1.69i)6-s + (−0.441 + 0.765i)7-s + 1.83·8-s + (0.857 − 0.514i)9-s − 1.90·10-s + (−0.150 + 0.261i)11-s + (−1.46 − 1.43i)12-s + (−0.653 − 1.13i)13-s + (−0.771 − 1.33i)14-s + (0.776 + 0.764i)15-s + (−0.574 + 0.994i)16-s − 0.778·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.357 - 0.934i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.357 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.490556 + 0.712859i\)
\(L(\frac12)\) \(\approx\) \(0.490556 + 0.712859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 + 0.461i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.23 - 2.13i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.21 - 2.10i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.16 - 2.02i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (2.35 + 4.07i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 - 7.77T + 19T^{2} \)
23 \( 1 + (1.37 + 2.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.18 + 2.05i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.685 - 1.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 + (1.77 + 3.07i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.73 + 6.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.103 - 0.180i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.11T + 53T^{2} \)
59 \( 1 + (-0.120 - 0.208i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.830 - 1.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.84 - 6.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.07T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + (6.35 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.25 - 9.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + (4.46 - 7.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.41716779824769410993948642092, −13.81389601399309085656520003739, −12.42239003264907429974602038613, −10.27472333327391457412285759835, −9.607742392770992060020545825915, −8.587047621713204809614361172403, −7.48286579933188513050371731775, −6.68449967536954070543052611740, −5.45880356007829472884828902541, −2.75932399467404698268855279009, 1.62424820920486031135675820796, 3.25578403595796048649443444071, 4.62671001098699171328357433170, 7.36475084377932927142074433104, 8.680484780498553269733903101776, 9.476971407994441382397427511475, 9.972933521363556965723913695087, 11.29548326812780193307231310377, 12.48319203581210370432692514123, 13.42454191780361585691677259630

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