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Properties

Label 2-92-23.9-c1-0-0
Degree $2$
Conductor $92$
Sign $0.869 - 0.493i$
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  + (1.89 + 2.19i)3-s + (−0.556 − 3.86i)5-s + (−1.06 + 2.32i)7-s + (−0.768 + 5.34i)9-s + (−2.52 − 0.740i)11-s + (−0.904 − 1.98i)13-s + (7.41 − 8.56i)15-s + (−0.0617 − 0.0396i)17-s + (0.698 − 0.448i)19-s + (−7.11 + 2.08i)21-s + (1.49 − 4.55i)23-s + (−9.85 + 2.89i)25-s + (−5.85 + 3.76i)27-s + (6.31 + 4.06i)29-s + (−4.12 + 4.75i)31-s + ⋯
L(s)  = 1  + (1.09 + 1.26i)3-s + (−0.248 − 1.73i)5-s + (−0.401 + 0.879i)7-s + (−0.256 + 1.78i)9-s + (−0.760 − 0.223i)11-s + (−0.250 − 0.549i)13-s + (1.91 − 2.21i)15-s + (−0.0149 − 0.00962i)17-s + (0.160 − 0.102i)19-s + (−1.55 + 0.456i)21-s + (0.312 − 0.949i)23-s + (−1.97 + 0.578i)25-s + (−1.12 + 0.723i)27-s + (1.17 + 0.754i)29-s + (−0.740 + 0.854i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.869 - 0.493i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 92,\ (\ :1/2),\ 0.869 - 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16399 + 0.306995i\)
\(L(\frac12)\) \(\approx\) \(1.16399 + 0.306995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.49 + 4.55i)T \)
good3 \( 1 + (-1.89 - 2.19i)T + (-0.426 + 2.96i)T^{2} \)
5 \( 1 + (0.556 + 3.86i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (1.06 - 2.32i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (2.52 + 0.740i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (0.904 + 1.98i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (0.0617 + 0.0396i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-0.698 + 0.448i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-6.31 - 4.06i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (4.12 - 4.75i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-0.834 + 5.80i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.336 - 2.34i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-5.43 - 6.27i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 4.84T + 47T^{2} \)
53 \( 1 + (-0.347 + 0.761i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (0.679 + 1.48i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (7.80 - 9.00i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-1.65 + 0.486i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (2.30 - 0.677i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-9.00 + 5.78i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-3.03 - 6.65i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-0.102 + 0.711i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-0.113 - 0.131i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (2.24 + 15.6i)T + (-93.0 + 27.3i)T^{2} \)
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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

−14.31687022823071820803189326712, −13.01575991722383143456136535991, −12.34614513031897125726141428718, −10.63191574936303038310175827432, −9.435137649634451867763163013449, −8.790597355927062754402848924054, −8.053094486670650645565230345928, −5.40533439964215059638747484880, −4.53053243782054121900328843158, −2.91650208968170250628165228900, 2.42830682292154710101942505897, 3.57041507171523526268934603683, 6.48833540363485734680428039114, 7.26768279940361326719009294049, 7.898011859329124275024836811409, 9.632989293819889628230476417178, 10.73565344286001388560380260501, 11.97550627019699408590801388159, 13.33971349160696780148049528502, 13.86652144825101062976829926149

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