Properties

Label 2-93-31.9-c1-0-2
Degree $2$
Conductor $93$
Sign $0.910 - 0.413i$
Analytic cond. $0.742608$
Root an. cond. $0.861747$
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  + (−0.102 + 0.315i)2-s + (0.978 + 0.207i)3-s + (1.52 + 1.11i)4-s + (−0.962 − 1.66i)5-s + (−0.165 + 0.287i)6-s + (−1.29 + 0.575i)7-s + (−1.04 + 0.757i)8-s + (0.913 + 0.406i)9-s + (0.624 − 0.132i)10-s + (0.138 − 1.32i)11-s + (1.26 + 1.40i)12-s + (−0.607 + 0.674i)13-s + (−0.0489 − 0.466i)14-s + (−0.595 − 1.83i)15-s + (1.03 + 3.18i)16-s + (−0.279 − 2.65i)17-s + ⋯
L(s)  = 1  + (−0.0724 + 0.222i)2-s + (0.564 + 0.120i)3-s + (0.764 + 0.555i)4-s + (−0.430 − 0.745i)5-s + (−0.0676 + 0.117i)6-s + (−0.488 + 0.217i)7-s + (−0.368 + 0.267i)8-s + (0.304 + 0.135i)9-s + (0.197 − 0.0419i)10-s + (0.0418 − 0.398i)11-s + (0.365 + 0.405i)12-s + (−0.168 + 0.186i)13-s + (−0.0130 − 0.124i)14-s + (−0.153 − 0.472i)15-s + (0.259 + 0.797i)16-s + (−0.0677 − 0.644i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.910 - 0.413i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.910 - 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11778 + 0.242183i\)
\(L(\frac12)\) \(\approx\) \(1.11778 + 0.242183i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-5.43 + 1.21i)T \)
good2 \( 1 + (0.102 - 0.315i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.962 + 1.66i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.29 - 0.575i)T + (4.68 - 5.20i)T^{2} \)
11 \( 1 + (-0.138 + 1.32i)T + (-10.7 - 2.28i)T^{2} \)
13 \( 1 + (0.607 - 0.674i)T + (-1.35 - 12.9i)T^{2} \)
17 \( 1 + (0.279 + 2.65i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (4.34 + 4.82i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (6.78 - 4.92i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.154 - 0.476i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (-0.749 + 1.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.14 + 1.30i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-2.70 - 3.00i)T + (-4.49 + 42.7i)T^{2} \)
47 \( 1 + (-4.08 - 12.5i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.23 - 1.43i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (-6.73 - 1.43i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + 2.87T + 61T^{2} \)
67 \( 1 + (3.11 + 5.39i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.94 - 2.64i)T + (47.5 + 52.7i)T^{2} \)
73 \( 1 + (-1.44 + 13.7i)T + (-71.4 - 15.1i)T^{2} \)
79 \( 1 + (-1.24 - 11.8i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (11.7 - 2.49i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 + (11.9 + 8.66i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-6.74 - 4.90i)T + (29.9 + 92.2i)T^{2} \)
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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

−14.13347848989969256153327710209, −12.94785625311874200212040460713, −12.10712368447201161551260857439, −11.06524191921918692856178150165, −9.465909784806480191124502701617, −8.484854539959438243505564875815, −7.51680421388645735487051825919, −6.17575079938870003238257600059, −4.27448577589092189126973408945, −2.68739475305117489728893768538, 2.31487124335254829254286461512, 3.83855942290851291273842506542, 6.14404833210602674836536989292, 7.07470334789051905499698087230, 8.312136050929198315345120913889, 10.02412263355705079180461246925, 10.49164220592043330231903875709, 11.84640622612417805941937867282, 12.78718418580445507047209400505, 14.28637464002864322839409108747

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