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Properties

Label 2-93-93.56-c2-0-2
Degree $2$
Conductor $93$
Sign $0.586 - 0.810i$
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  + 0.649i·2-s + (−2.99 − 0.133i)3-s + 3.57·4-s + (0.102 + 0.0591i)5-s + (0.0869 − 1.94i)6-s + (3.56 + 6.17i)7-s + 4.92i·8-s + (8.96 + 0.802i)9-s + (−0.0384 + 0.0665i)10-s + (8.20 + 4.73i)11-s + (−10.7 − 0.478i)12-s + (−4.04 + 7.01i)13-s + (−4.01 + 2.31i)14-s + (−0.298 − 0.190i)15-s + 11.1·16-s + (−1.93 + 1.11i)17-s + ⋯
L(s)  = 1  + 0.324i·2-s + (−0.999 − 0.0446i)3-s + 0.894·4-s + (0.0204 + 0.0118i)5-s + (0.0144 − 0.324i)6-s + (0.509 + 0.882i)7-s + 0.615i·8-s + (0.996 + 0.0891i)9-s + (−0.00384 + 0.00665i)10-s + (0.745 + 0.430i)11-s + (−0.893 − 0.0399i)12-s + (−0.311 + 0.539i)13-s + (−0.286 + 0.165i)14-s + (−0.0199 − 0.0127i)15-s + 0.694·16-s + (−0.114 + 0.0658i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.586 - 0.810i$
Analytic conductor: \(2.53406\)
Root analytic conductor: \(1.59187\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1),\ 0.586 - 0.810i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.11817 + 0.571260i\)
\(L(\frac12)\) \(\approx\) \(1.11817 + 0.571260i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.99 + 0.133i)T \)
31 \( 1 + (-1.99 - 30.9i)T \)
good2 \( 1 - 0.649iT - 4T^{2} \)
5 \( 1 + (-0.102 - 0.0591i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.56 - 6.17i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-8.20 - 4.73i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (4.04 - 7.01i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (1.93 - 1.11i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (0.396 + 0.685i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + 21.8iT - 529T^{2} \)
29 \( 1 + 52.5iT - 841T^{2} \)
37 \( 1 + (-4.03 - 6.98i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (12.3 + 7.12i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (5.46 + 9.45i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 63.1iT - 2.20e3T^{2} \)
53 \( 1 + (53.9 + 31.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-14.6 + 8.45i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + 52.8T + 3.72e3T^{2} \)
67 \( 1 + (-47.7 + 82.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-112. - 65.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-17.0 + 29.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (34.0 + 58.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (11.4 + 6.59i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 30.1iT - 7.92e3T^{2} \)
97 \( 1 + 74.7T + 9.40e3T^{2} \)
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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

−14.19023553181883145502685368271, −12.39151985458042255616498065555, −11.87532926069406443182496649482, −11.00946962066775772784648220774, −9.744818542719794396547568632940, −8.158087841628006085090790728637, −6.80002732902199165476726114587, −6.00714247924385000420124075465, −4.63299159300331274327913038283, −2.03200229358840483150408340612, 1.31559842076696099533044583975, 3.74196367986322659664116748433, 5.42196345747902602762097034570, 6.73754855886146984750792569211, 7.64622313892386683108465735118, 9.655890364335342571963556222446, 10.78141645179451123634026648972, 11.32479056099589166531076562109, 12.28326136768289483922204401622, 13.41651939922228017195980521321

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