↑

Properties

Label 2-92-23.9-c1-0-1
Degree $2$
Conductor $92$
Sign $0.135 + 0.990i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 − 1.56i)3-s + (−0.187 − 1.30i)5-s + (1.18 − 2.58i)7-s + (−0.181 + 1.26i)9-s + (−2.10 − 0.617i)11-s + (2.61 + 5.73i)13-s + (−1.78 + 2.06i)15-s + (2.92 + 1.88i)17-s + (0.593 − 0.381i)19-s + (−5.64 + 1.65i)21-s + (3.67 + 3.07i)23-s + (3.12 − 0.917i)25-s + (−3.00 + 1.92i)27-s + (−7.98 − 5.13i)29-s + (2.11 − 2.44i)31-s + ⋯
L(s)  = 1  + (−0.781 − 0.902i)3-s + (−0.0840 − 0.584i)5-s + (0.446 − 0.977i)7-s + (−0.0604 + 0.420i)9-s + (−0.633 − 0.186i)11-s + (0.725 + 1.58i)13-s + (−0.461 + 0.532i)15-s + (0.709 + 0.456i)17-s + (0.136 − 0.0875i)19-s + (−1.23 + 0.361i)21-s + (0.766 + 0.642i)23-s + (0.624 − 0.183i)25-s + (−0.577 + 0.371i)27-s + (−1.48 − 0.953i)29-s + (0.380 − 0.439i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.135 + 0.990i$
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: \((2,\ 92,\ (\ :1/2),\ 0.135 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603755 - 0.526538i\)
\(L(\frac12)\) \(\approx\) \(0.603755 - 0.526538i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.67 - 3.07i)T \)
good3 \( 1 + (1.35 + 1.56i)T + (-0.426 + 2.96i)T^{2} \)
5 \( 1 + (0.187 + 1.30i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (-1.18 + 2.58i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (2.10 + 0.617i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-2.61 - 5.73i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-2.92 - 1.88i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-0.593 + 0.381i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (7.98 + 5.13i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-2.11 + 2.44i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-0.667 + 4.64i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.422 - 2.93i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-7.93 - 9.16i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 7.35T + 47T^{2} \)
53 \( 1 + (-4.41 + 9.66i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-0.411 - 0.900i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (3.23 - 3.73i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (2.10 - 0.616i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (8.62 - 2.53i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (7.78 - 5.00i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-1.67 - 3.66i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (2.38 - 16.5i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-1.78 - 2.06i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (0.835 + 5.80i)T + (-93.0 + 27.3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.49933072183521058640769775196, −12.87393910817010649857479386857, −11.61766025704914107481491735120, −11.00957956452597868988828091154, −9.415130736035718756597726998373, −7.961396107900290882577466039570, −6.96456262710522309813809456386, −5.71041776952781010341969024456, −4.18820124433950787586993775112, −1.28018652023636663535204959689, 3.09436552775246597526899220966, 5.05865861129112002113073092789, 5.76081423949302956430942384213, 7.57035671949913388210408670857, 8.907103089074253784711952521699, 10.41026316974869428102387549487, 10.82316628703672785411107561956, 11.99416862715317576115903302601, 13.12027445890396814481764715192, 14.70902582980554499242524243364

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