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Properties

Label 2-93-93.35-c2-0-13
Degree $2$
Conductor $93$
Sign $-0.394 + 0.918i$
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  + (−3.46 + 1.12i)2-s + (2.05 − 2.18i)3-s + (7.51 − 5.46i)4-s − 5.25i·5-s + (−4.65 + 9.89i)6-s + (−8.69 + 6.31i)7-s + (−11.3 + 15.6i)8-s + (−0.568 − 8.98i)9-s + (5.92 + 18.2i)10-s + (−5.73 − 7.89i)11-s + (3.48 − 27.6i)12-s + (2.39 − 7.36i)13-s + (23.0 − 31.6i)14-s + (−11.5 − 10.7i)15-s + (10.2 − 31.5i)16-s + (−11.7 + 16.1i)17-s + ⋯
L(s)  = 1  + (−1.73 + 0.563i)2-s + (0.684 − 0.729i)3-s + (1.87 − 1.36i)4-s − 1.05i·5-s + (−0.775 + 1.64i)6-s + (−1.24 + 0.902i)7-s + (−1.41 + 1.95i)8-s + (−0.0631 − 0.998i)9-s + (0.592 + 1.82i)10-s + (−0.521 − 0.717i)11-s + (0.290 − 2.30i)12-s + (0.183 − 0.566i)13-s + (1.64 − 2.26i)14-s + (−0.766 − 0.719i)15-s + (0.640 − 1.97i)16-s + (−0.689 + 0.949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.394 + 0.918i$
Analytic conductor: \(2.53406\)
Root analytic conductor: \(1.59187\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1),\ -0.394 + 0.918i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.262107 - 0.397845i\)
\(L(\frac12)\) \(\approx\) \(0.262107 - 0.397845i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.05 + 2.18i)T \)
31 \( 1 + (30.9 - 2.21i)T \)
good2 \( 1 + (3.46 - 1.12i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + 5.25iT - 25T^{2} \)
7 \( 1 + (8.69 - 6.31i)T + (15.1 - 46.6i)T^{2} \)
11 \( 1 + (5.73 + 7.89i)T + (-37.3 + 115. i)T^{2} \)
13 \( 1 + (-2.39 + 7.36i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (11.7 - 16.1i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (8.21 + 25.2i)T + (-292. + 212. i)T^{2} \)
23 \( 1 + (-11.4 + 15.7i)T + (-163. - 503. i)T^{2} \)
29 \( 1 + (-18.2 + 5.91i)T + (680. - 494. i)T^{2} \)
37 \( 1 - 36.1T + 1.36e3T^{2} \)
41 \( 1 + (10.7 - 3.48i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + (-3.28 - 10.1i)T + (-1.49e3 + 1.08e3i)T^{2} \)
47 \( 1 + (-46.0 - 14.9i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-8.46 + 11.6i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (-0.297 - 0.0965i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 - 91.8T + 3.72e3T^{2} \)
67 \( 1 - 38.4T + 4.48e3T^{2} \)
71 \( 1 + (-13.0 + 18.0i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-37.0 + 26.9i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (24.6 + 17.8i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (113. - 36.7i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + (-67.0 - 92.2i)T + (-2.44e3 + 7.53e3i)T^{2} \)
97 \( 1 + (108. - 79.0i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−13.17572125273320047070296144467, −12.59914239987942792043318192800, −10.98053468659160572988734130653, −9.561692386147875922865106291375, −8.728914241993623365390264296663, −8.368204552857308568281812696430, −6.84051280812648600851877087561, −5.88590432944382467535659398192, −2.57261439996693944453995333916, −0.56645877842123159328415361902, 2.48357643389020531142267334205, 3.67410460090234480711733161560, 6.84005359709915601484306601811, 7.54921977055727351932060893799, 9.000811558849216521137949614769, 9.928915913301108915875131176293, 10.38862286561285101099258513693, 11.31866451730109354409455638682, 12.95051273467002371256815037048, 14.25730948862471606363014180072

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