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Properties

Label 2-93-31.2-c1-0-3
Degree $2$
Conductor $93$
Sign $0.819 - 0.572i$
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  + (1.47 + 1.07i)2-s + (0.809 − 0.587i)3-s + (0.413 + 1.27i)4-s − 2.95·5-s + 1.82·6-s + (0.795 + 2.44i)7-s + (0.373 − 1.14i)8-s + (0.309 − 0.951i)9-s + (−4.36 − 3.17i)10-s + (−1.27 − 3.91i)11-s + (1.08 + 0.786i)12-s + (−4.38 + 3.18i)13-s + (−1.45 + 4.47i)14-s + (−2.39 + 1.73i)15-s + (3.95 − 2.87i)16-s + (0.551 − 1.69i)17-s + ⋯
L(s)  = 1  + (1.04 + 0.759i)2-s + (0.467 − 0.339i)3-s + (0.206 + 0.636i)4-s − 1.32·5-s + 0.745·6-s + (0.300 + 0.925i)7-s + (0.132 − 0.406i)8-s + (0.103 − 0.317i)9-s + (−1.38 − 1.00i)10-s + (−0.384 − 1.18i)11-s + (0.312 + 0.227i)12-s + (−1.21 + 0.884i)13-s + (−0.388 + 1.19i)14-s + (−0.617 + 0.448i)15-s + (0.988 − 0.717i)16-s + (0.133 − 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.819 - 0.572i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.819 - 0.572i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45064 + 0.456577i\)
\(L(\frac12)\) \(\approx\) \(1.45064 + 0.456577i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.51 + 3.25i)T \)
good2 \( 1 + (-1.47 - 1.07i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + 2.95T + 5T^{2} \)
7 \( 1 + (-0.795 - 2.44i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (1.27 + 3.91i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (4.38 - 3.18i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.551 + 1.69i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.89 - 2.10i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.79 - 5.52i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.982 - 0.713i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 4.21T + 37T^{2} \)
41 \( 1 + (-7.70 - 5.59i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.74 + 3.44i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (7.33 - 5.32i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.11 + 9.58i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.27 - 5.28i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 7.32T + 61T^{2} \)
67 \( 1 + 6.47T + 67T^{2} \)
71 \( 1 + (-0.597 + 1.83i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.315 - 0.971i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.78 + 8.57i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.42 + 3.93i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.88 + 8.88i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.73 - 8.42i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.29943128612193302125430213442, −13.35689128800543617848724049822, −12.07370842055992334588221331152, −11.60117493566773050068372769313, −9.550726686211190036335301273952, −8.102003602303747675787599852741, −7.37931631289131877265105803798, −5.91104471333643364421711144006, −4.61570760961902014241942432366, −3.19620498380622204943100560952, 2.82132060776725685087116602279, 4.20153806672991728537985433057, 4.83399862836563314021517085059, 7.41341216710461944808978739430, 8.078948895112005659606841090159, 10.04559982985428459190442522445, 10.88123308311728605358496531145, 12.10295025592644950230750175173, 12.67932913143295849579239721256, 13.91969721935894180943936144802

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