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Properties

Label 2-93-31.19-c1-0-3
Degree $2$
Conductor $93$
Sign $0.906 - 0.421i$
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  + (0.927 + 0.673i)2-s + (0.913 + 0.406i)3-s + (−0.211 − 0.651i)4-s + (0.430 − 0.746i)5-s + (0.573 + 0.993i)6-s + (−2.45 + 2.72i)7-s + (0.951 − 2.92i)8-s + (0.669 + 0.743i)9-s + (0.902 − 0.401i)10-s + (−5.09 − 1.08i)11-s + (0.0716 − 0.681i)12-s + (0.427 + 4.06i)13-s + (−4.11 + 0.874i)14-s + (0.696 − 0.506i)15-s + (1.74 − 1.26i)16-s + (2.06 − 0.439i)17-s + ⋯
L(s)  = 1  + (0.655 + 0.476i)2-s + (0.527 + 0.234i)3-s + (−0.105 − 0.325i)4-s + (0.192 − 0.333i)5-s + (0.234 + 0.405i)6-s + (−0.927 + 1.03i)7-s + (0.336 − 1.03i)8-s + (0.223 + 0.247i)9-s + (0.285 − 0.127i)10-s + (−1.53 − 0.326i)11-s + (0.0206 − 0.196i)12-s + (0.118 + 1.12i)13-s + (−1.09 + 0.233i)14-s + (0.179 − 0.130i)15-s + (0.436 − 0.317i)16-s + (0.501 − 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.906 - 0.421i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.906 - 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35467 + 0.299550i\)
\(L(\frac12)\) \(\approx\) \(1.35467 + 0.299550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 - 0.406i)T \)
31 \( 1 + (-0.955 - 5.48i)T \)
good2 \( 1 + (-0.927 - 0.673i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-0.430 + 0.746i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.45 - 2.72i)T + (-0.731 - 6.96i)T^{2} \)
11 \( 1 + (5.09 + 1.08i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.427 - 4.06i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (-2.06 + 0.439i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-0.612 + 5.82i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (-1.39 + 4.30i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.03 + 0.748i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (1.20 + 2.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.45 + 1.53i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (0.814 - 7.74i)T + (-42.0 - 8.94i)T^{2} \)
47 \( 1 + (-4.60 + 3.34i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-9.00 - 10.0i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (7.65 + 3.40i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + 2.01T + 61T^{2} \)
67 \( 1 + (-4.11 + 7.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.187 + 0.208i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (2.52 + 0.537i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (4.45 - 0.946i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (7.76 - 3.45i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (1.07 + 3.32i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.49 + 16.9i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.07407456187537893300512535270, −13.23780871157411302993105866795, −12.50490809976176526155274016625, −10.76648803302982707736195947477, −9.535407596120452415636410401525, −8.804494031439580383746582953180, −7.05928557712378011634056797425, −5.76547653727984735564592303751, −4.74855206246522363573755137642, −2.86442373439229363259420479728, 2.78270807065156215314616827879, 3.79451069419511428713258422818, 5.56608821261703228609444749636, 7.38458599326645327686721156163, 8.118596603196932849436508706692, 9.989000024169060584088638815916, 10.61602736062889286045419775791, 12.27356500419190815767792259396, 13.11510267822203793502520063444, 13.55976148282072027485778389011

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