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Properties

Label 2-93-31.23-c2-0-5
Degree $2$
Conductor $93$
Sign $0.936 - 0.350i$
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  + (−0.103 + 0.317i)2-s + (1.64 − 0.535i)3-s + (3.14 + 2.28i)4-s + 1.90·5-s + 0.578i·6-s + (−2.74 − 1.99i)7-s + (−2.13 + 1.54i)8-s + (2.42 − 1.76i)9-s + (−0.196 + 0.604i)10-s + (−6.79 + 9.35i)11-s + (6.40 + 2.08i)12-s + (22.9 − 7.46i)13-s + (0.917 − 0.666i)14-s + (3.13 − 1.01i)15-s + (4.53 + 13.9i)16-s + (−17.3 − 23.9i)17-s + ⋯
L(s)  = 1  + (−0.0516 + 0.158i)2-s + (0.549 − 0.178i)3-s + (0.786 + 0.571i)4-s + 0.380·5-s + 0.0964i·6-s + (−0.392 − 0.285i)7-s + (−0.266 + 0.193i)8-s + (0.269 − 0.195i)9-s + (−0.0196 + 0.0604i)10-s + (−0.618 + 0.850i)11-s + (0.533 + 0.173i)12-s + (1.76 − 0.574i)13-s + (0.0655 − 0.0476i)14-s + (0.208 − 0.0678i)15-s + (0.283 + 0.872i)16-s + (−1.02 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.936 - 0.350i$
Analytic conductor: \(2.53406\)
Root analytic conductor: \(1.59187\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1),\ 0.936 - 0.350i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.70867 + 0.308858i\)
\(L(\frac12)\) \(\approx\) \(1.70867 + 0.308858i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 + 0.535i)T \)
31 \( 1 + (2.97 + 30.8i)T \)
good2 \( 1 + (0.103 - 0.317i)T + (-3.23 - 2.35i)T^{2} \)
5 \( 1 - 1.90T + 25T^{2} \)
7 \( 1 + (2.74 + 1.99i)T + (15.1 + 46.6i)T^{2} \)
11 \( 1 + (6.79 - 9.35i)T + (-37.3 - 115. i)T^{2} \)
13 \( 1 + (-22.9 + 7.46i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (17.3 + 23.9i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (4.03 - 12.4i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + (-0.955 - 1.31i)T + (-163. + 503. i)T^{2} \)
29 \( 1 + (43.6 + 14.1i)T + (680. + 494. i)T^{2} \)
37 \( 1 - 22.1iT - 1.36e3T^{2} \)
41 \( 1 + (-4.11 + 12.6i)T + (-1.35e3 - 988. i)T^{2} \)
43 \( 1 + (3.04 + 0.989i)T + (1.49e3 + 1.08e3i)T^{2} \)
47 \( 1 + (2.40 + 7.40i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-45.0 - 62.0i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (-30.8 - 95.0i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + 36.6iT - 3.72e3T^{2} \)
67 \( 1 - 4.14T + 4.48e3T^{2} \)
71 \( 1 + (-38.3 + 27.8i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (0.210 - 0.289i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-34.9 - 48.0i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-99.2 - 32.2i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + (-65.1 + 89.6i)T + (-2.44e3 - 7.53e3i)T^{2} \)
97 \( 1 + (49.3 + 35.8i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−13.49095146755154325405664756599, −13.13893020617856287769998813630, −11.74615958246898353765122869642, −10.64340890501501624055724480018, −9.378653874483236329154877556220, −8.097961709635185356041111564264, −7.14577987733349293822308832669, −5.93994923836453064040393340289, −3.76124932078243775483507628797, −2.25061734166369204385469686602, 1.93591122011117849335736348695, 3.57096821495414635908382462866, 5.74221174244416123612682152372, 6.62764420825319990679413607104, 8.385907161956128413812437300868, 9.318314212625518855173573961394, 10.70801735941920244082266291835, 11.18697364623173922279866135834, 12.91729183647485403907629795057, 13.66360397372719592512360362804

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