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Properties

Label 2-93-93.5-c2-0-14
Degree $2$
Conductor $93$
Sign $-0.0785 + 0.996i$
Analytic cond. $2.53406$
Root an. cond. $1.59187$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.32i·2-s + (2.99 − 0.196i)3-s − 1.42·4-s + (−3.71 + 2.14i)5-s + (−0.456 − 6.97i)6-s + (4.74 − 8.20i)7-s − 6.00i·8-s + (8.92 − 1.17i)9-s + (5.00 + 8.66i)10-s + (−10.0 + 5.79i)11-s + (−4.26 + 0.278i)12-s + (5.91 + 10.2i)13-s + (−19.1 − 11.0i)14-s + (−10.7 + 7.15i)15-s − 19.6·16-s + (6.92 + 3.99i)17-s + ⋯
L(s)  = 1  − 1.16i·2-s + (0.997 − 0.0653i)3-s − 0.355·4-s + (−0.743 + 0.429i)5-s + (−0.0760 − 1.16i)6-s + (0.677 − 1.17i)7-s − 0.750i·8-s + (0.991 − 0.130i)9-s + (0.500 + 0.866i)10-s + (−0.912 + 0.526i)11-s + (−0.355 + 0.0232i)12-s + (0.455 + 0.788i)13-s + (−1.36 − 0.788i)14-s + (−0.714 + 0.477i)15-s − 1.22·16-s + (0.407 + 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0785 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0785 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.0785 + 0.996i$
Analytic conductor: \(2.53406\)
Root analytic conductor: \(1.59187\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1),\ -0.0785 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.17605 - 1.27231i\)
\(L(\frac12)\) \(\approx\) \(1.17605 - 1.27231i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.99 + 0.196i)T \)
31 \( 1 + (22.0 - 21.8i)T \)
good2 \( 1 + 2.32iT - 4T^{2} \)
5 \( 1 + (3.71 - 2.14i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-4.74 + 8.20i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (10.0 - 5.79i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.91 - 10.2i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-6.92 - 3.99i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (1.89 - 3.28i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 - 20.8iT - 529T^{2} \)
29 \( 1 - 11.4iT - 841T^{2} \)
37 \( 1 + (-21.5 + 37.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-62.6 + 36.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (32.4 - 56.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 62.9iT - 2.20e3T^{2} \)
53 \( 1 + (4.06 - 2.34i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (64.5 + 37.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + 110.T + 3.72e3T^{2} \)
67 \( 1 + (7.88 + 13.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (10.1 - 5.86i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-39.4 - 68.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-58.7 + 101. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (55.4 - 31.9i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 24.7iT - 7.92e3T^{2} \)
97 \( 1 - 129.T + 9.40e3T^{2} \)
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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

−13.41169192967363031784775610221, −12.42972739026004702727019303919, −11.15914807897987674952398120820, −10.53659321932967430915112494042, −9.383929888006882777774043384710, −7.78898517633848028453948197295, −7.15823671840647989324822915714, −4.29098101016307586569554892322, −3.37576009714839961759830924010, −1.68079762420961808252089935910, 2.67832748947145189196946921784, 4.72833579672749546934583332694, 5.96185753275512273108527240631, 7.84975440866464234367111826525, 8.079680565510542713168417428988, 9.059730304118401867887472478749, 10.85316395910048440647411616125, 12.12494957087625977710673487104, 13.30259752591644894615621535977, 14.47670541698653607787005521492

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