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Properties

Label 2-92-23.13-c1-0-0
Degree $2$
Conductor $92$
Sign $0.892 - 0.451i$
Analytic cond. $0.734623$
Root an. cond. $0.857101$
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  + (1.98 + 1.27i)3-s + (−0.105 + 0.230i)5-s + (−3.93 − 1.15i)7-s + (1.06 + 2.32i)9-s + (1.21 − 1.40i)11-s + (2.00 − 0.589i)13-s + (−0.502 + 0.323i)15-s + (−0.898 − 6.25i)17-s + (−0.956 + 6.65i)19-s + (−6.32 − 7.29i)21-s + (−4.56 + 1.48i)23-s + (3.23 + 3.73i)25-s + (0.148 − 1.03i)27-s + (−0.592 − 4.12i)29-s + (−5.71 + 3.67i)31-s + ⋯
L(s)  = 1  + (1.14 + 0.735i)3-s + (−0.0471 + 0.103i)5-s + (−1.48 − 0.436i)7-s + (0.353 + 0.775i)9-s + (0.366 − 0.422i)11-s + (0.556 − 0.163i)13-s + (−0.129 + 0.0834i)15-s + (−0.217 − 1.51i)17-s + (−0.219 + 1.52i)19-s + (−1.38 − 1.59i)21-s + (−0.951 + 0.308i)23-s + (0.646 + 0.746i)25-s + (0.0286 − 0.199i)27-s + (−0.110 − 0.765i)29-s + (−1.02 + 0.660i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.892 - 0.451i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 92,\ (\ :1/2),\ 0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17378 + 0.280059i\)
\(L(\frac12)\) \(\approx\) \(1.17378 + 0.280059i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + (4.56 - 1.48i)T \)
good3 \( 1 + (-1.98 - 1.27i)T + (1.24 + 2.72i)T^{2} \)
5 \( 1 + (0.105 - 0.230i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (3.93 + 1.15i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-1.21 + 1.40i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-2.00 + 0.589i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (0.898 + 6.25i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.956 - 6.65i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.592 + 4.12i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (5.71 - 3.67i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-0.771 - 1.68i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (1.82 - 3.98i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-1.87 - 1.20i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 - 13.6T + 47T^{2} \)
53 \( 1 + (-7.33 - 2.15i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (10.1 - 2.97i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (4.37 - 2.81i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (2.47 + 2.85i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (0.813 + 0.938i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.676 + 4.70i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-3.53 + 1.03i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (4.20 + 9.20i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-11.1 - 7.14i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (3.26 - 7.15i)T + (-63.5 - 73.3i)T^{2} \)
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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

−14.04124792722453566723626009122, −13.47470462136795067272156605556, −12.13378294411289948356493079735, −10.56823220901941325737953851307, −9.628891144467328089670108380858, −8.910969322496695504257812077509, −7.50094168464691701622445700203, −6.06193340576438234627856635594, −3.94254543242120956861776676779, −3.08828830180894641792282261638, 2.35401475583982768211390930939, 3.80512979761728211225752170457, 6.17469396828647115041193901533, 7.15966181253812145399046312241, 8.629150391542818270241051248309, 9.206441174876411516793837806169, 10.63688627236832441458322967146, 12.40811399518071087538394598468, 12.92242326001223858342467130627, 13.83632659491507195469580907626

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