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Properties

Label 2-93-93.77-c1-0-1
Degree $2$
Conductor $93$
Sign $-0.211 - 0.977i$
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  + (1.43 + 1.98i)2-s + (−1.57 − 0.720i)3-s + (−1.23 + 3.79i)4-s + 2.68i·5-s + (−0.840 − 4.15i)6-s + (1.40 − 4.31i)7-s + (−4.64 + 1.50i)8-s + (1.96 + 2.26i)9-s + (−5.32 + 3.86i)10-s + (0.328 − 1.01i)11-s + (4.68 − 5.09i)12-s + (0.720 − 0.992i)13-s + (10.5 − 3.43i)14-s + (1.93 − 4.23i)15-s + (−3.21 − 2.33i)16-s + (−1.20 − 3.69i)17-s + ⋯
L(s)  = 1  + (1.01 + 1.40i)2-s + (−0.909 − 0.415i)3-s + (−0.617 + 1.89i)4-s + 1.20i·5-s + (−0.343 − 1.69i)6-s + (0.530 − 1.63i)7-s + (−1.64 + 0.533i)8-s + (0.654 + 0.756i)9-s + (−1.68 + 1.22i)10-s + (0.0990 − 0.304i)11-s + (1.35 − 1.47i)12-s + (0.199 − 0.275i)13-s + (2.82 − 0.917i)14-s + (0.499 − 1.09i)15-s + (−0.802 − 0.583i)16-s + (−0.291 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.211 - 0.977i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ -0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.786413 + 0.974464i\)
\(L(\frac12)\) \(\approx\) \(0.786413 + 0.974464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.57 + 0.720i)T \)
31 \( 1 + (4.53 - 3.22i)T \)
good2 \( 1 + (-1.43 - 1.98i)T + (-0.618 + 1.90i)T^{2} \)
5 \( 1 - 2.68iT - 5T^{2} \)
7 \( 1 + (-1.40 + 4.31i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.328 + 1.01i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.720 + 0.992i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.20 + 3.69i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.60 - 1.89i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.773 + 2.37i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.50 - 1.82i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 4.63iT - 37T^{2} \)
41 \( 1 + (4.50 + 6.19i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-1.32 - 1.82i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-0.902 + 1.24i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.09 - 6.45i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.580 + 0.798i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + 12.1iT - 61T^{2} \)
67 \( 1 - 2.90T + 67T^{2} \)
71 \( 1 + (-9.48 + 3.08i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.79 - 2.53i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.953 + 0.309i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.48 - 2.52i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (5.20 - 16.0i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.0266 + 0.0820i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.13956353462847427167132287419, −13.75838953352514913663349803184, −12.61032985410520529086450282069, −11.15702053363605512075188402150, −10.46907661320536988089669252309, −7.949132951161606468407157171918, −7.05660802161303893432183618953, −6.52986185885625124796399724492, −5.10328893616936945469467640075, −3.81613092128111139010885988950, 1.86297130808559404177438537732, 4.13419350022149562293070667136, 5.12607225244482918154819440254, 5.89354136536102048753313170244, 8.732423037172346099869717655032, 9.649666779449857248345101041431, 11.05687035591574693864209742910, 11.75210883083523625669056797716, 12.53114948499371687258160733183, 13.07887324335058707726760166030

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