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Properties

Label 2-99-33.2-c1-0-2
Degree $2$
Conductor $99$
Sign $0.937 + 0.347i$
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  + (0.212 − 0.654i)2-s + (1.23 + 0.897i)4-s + (−0.0381 + 0.0123i)5-s + (0.145 − 0.199i)7-s + (1.96 − 1.42i)8-s + 0.0276i·10-s + (−3.12 − 1.12i)11-s + (−2.18 − 0.711i)13-s + (−0.0998 − 0.137i)14-s + (0.427 + 1.31i)16-s + (1.32 + 4.06i)17-s + (−3.64 − 5.01i)19-s + (−0.0582 − 0.0189i)20-s + (−1.39 + 1.80i)22-s + 6.79i·23-s + ⋯
L(s)  = 1  + (0.150 − 0.462i)2-s + (0.617 + 0.448i)4-s + (−0.0170 + 0.00554i)5-s + (0.0548 − 0.0754i)7-s + (0.694 − 0.504i)8-s + 0.00872i·10-s + (−0.940 − 0.338i)11-s + (−0.607 − 0.197i)13-s + (−0.0266 − 0.0367i)14-s + (0.106 + 0.328i)16-s + (0.320 + 0.986i)17-s + (−0.836 − 1.15i)19-s + (−0.0130 − 0.00422i)20-s + (−0.298 + 0.384i)22-s + 1.41i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.937 + 0.347i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.937 + 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16530 - 0.208991i\)
\(L(\frac12)\) \(\approx\) \(1.16530 - 0.208991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (3.12 + 1.12i)T \)
good2 \( 1 + (-0.212 + 0.654i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.0381 - 0.0123i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.145 + 0.199i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.18 + 0.711i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.32 - 4.06i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.64 + 5.01i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.79iT - 23T^{2} \)
29 \( 1 + (4.52 + 3.28i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.48 + 4.56i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.26 - 2.36i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.76 + 5.64i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.03iT - 43T^{2} \)
47 \( 1 + (6.53 + 9.00i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-8.52 - 2.77i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.63 + 2.25i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-8.06 + 2.62i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 7.94T + 67T^{2} \)
71 \( 1 + (3.16 - 1.02i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.96 - 9.58i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.86 + 0.930i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.63 - 5.03i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.54iT - 89T^{2} \)
97 \( 1 + (0.935 - 2.88i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.41507781389244600391965889519, −12.88058319647035432057816197594, −11.62970319781068613558543279928, −10.87304424456861432086090963913, −9.766680998953859395729165377320, −8.132085576066644188936212241408, −7.23993775784516113832692016751, −5.67100621532182790986433515438, −3.90525435360678872864559383888, −2.37075233636542388048259931911, 2.38550035670366967048423833177, 4.71345013764879686427269231148, 5.92053789021530430161471007298, 7.15997481036572300016606830833, 8.156945217958100543031608780677, 9.844158331593470205139800310681, 10.67698248892952955004053609997, 11.88385801453708870404574428996, 12.93258120675697026666191176040, 14.36692593133079599914773247708

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