Properties

Label 2-93-31.16-c1-0-5
Degree $2$
Conductor $93$
Sign $0.371 + 0.928i$
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  + (1.72 − 1.25i)2-s + (−0.809 − 0.587i)3-s + (0.784 − 2.41i)4-s − 0.560·5-s − 2.13·6-s + (−0.673 + 2.07i)7-s + (−0.355 − 1.09i)8-s + (0.309 + 0.951i)9-s + (−0.965 + 0.701i)10-s + (0.294 − 0.905i)11-s + (−2.05 + 1.49i)12-s + (1.51 + 1.10i)13-s + (1.43 + 4.41i)14-s + (0.453 + 0.329i)15-s + (2.12 + 1.54i)16-s + (−2.08 − 6.42i)17-s + ⋯
L(s)  = 1  + (1.21 − 0.885i)2-s + (−0.467 − 0.339i)3-s + (0.392 − 1.20i)4-s − 0.250·5-s − 0.869·6-s + (−0.254 + 0.783i)7-s + (−0.125 − 0.386i)8-s + (0.103 + 0.317i)9-s + (−0.305 + 0.221i)10-s + (0.0886 − 0.272i)11-s + (−0.593 + 0.431i)12-s + (0.420 + 0.305i)13-s + (0.383 + 1.17i)14-s + (0.117 + 0.0850i)15-s + (0.531 + 0.386i)16-s + (−0.506 − 1.55i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.371 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20836 - 0.818371i\)
\(L(\frac12)\) \(\approx\) \(1.20836 - 0.818371i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (1.98 + 5.20i)T \)
good2 \( 1 + (-1.72 + 1.25i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + 0.560T + 5T^{2} \)
7 \( 1 + (0.673 - 2.07i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.294 + 0.905i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.51 - 1.10i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.08 + 6.42i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.62 - 4.08i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.574 - 1.76i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-2.99 + 2.17i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 5.88T + 37T^{2} \)
41 \( 1 + (-0.965 + 0.701i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (4.76 - 3.46i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (8.48 + 6.16i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.96 + 9.13i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-10.4 - 7.62i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 6.37T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + (-0.308 - 0.948i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.189 + 0.584i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.20 - 3.71i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (11.3 - 8.27i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.31 - 10.1i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.88 + 11.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.49381662884558760590201006300, −12.77925076923043998362780188078, −11.65720533816879263169634653449, −11.36222878828826775325512275724, −9.835482831095833881093502648067, −8.254350321608956669717088217487, −6.45160874484929622163076402351, −5.37931796189446771148950461021, −3.99579629814104825409478506756, −2.33711861705479428002979827378, 3.76898978213619365036689760161, 4.65355181016421613409941022710, 6.13436838951947286276339633040, 6.92003766954346452975991907098, 8.370146104954352759630763986671, 10.15899129359338020598890212472, 11.13877662069678911497736011411, 12.64865900374087038447006598778, 13.15546594075431595829691537769, 14.39900795783977117344873811383

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