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Properties

Label 2-2169-241.26-c0-0-0
Degree $2$
Conductor $2169$
Sign $0.860 - 0.508i$
Analytic cond. $1.08247$
Root an. cond. $1.04041$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)4-s + (1.74 − 0.0685i)7-s + (0.0638 + 0.226i)13-s + 1.00i·16-s + (−0.532 + 0.105i)19-s + (0.156 − 0.987i)25-s + (1.28 + 1.18i)28-s + (−1.04 − 0.384i)31-s + (−1.41 − 0.398i)37-s + (−0.568 − 0.614i)43-s + (2.03 − 0.160i)49-s + (−0.114 + 0.205i)52-s + (0.0245 + 0.311i)61-s + (−0.707 + 0.707i)64-s + (−0.882 + 0.211i)67-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)4-s + (1.74 − 0.0685i)7-s + (0.0638 + 0.226i)13-s + 1.00i·16-s + (−0.532 + 0.105i)19-s + (0.156 − 0.987i)25-s + (1.28 + 1.18i)28-s + (−1.04 − 0.384i)31-s + (−1.41 − 0.398i)37-s + (−0.568 − 0.614i)43-s + (2.03 − 0.160i)49-s + (−0.114 + 0.205i)52-s + (0.0245 + 0.311i)61-s + (−0.707 + 0.707i)64-s + (−0.882 + 0.211i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2169\)    =    \(3^{2} \cdot 241\)
Sign: $0.860 - 0.508i$
Analytic conductor: \(1.08247\)
Root analytic conductor: \(1.04041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2169} (1954, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2169,\ (\ :0),\ 0.860 - 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.641384793\)
\(L(\frac12)\) \(\approx\) \(1.641384793\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
241 \( 1 + (0.760 + 0.649i)T \)
good2 \( 1 + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (-0.156 + 0.987i)T^{2} \)
7 \( 1 + (-1.74 + 0.0685i)T + (0.996 - 0.0784i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + (-0.0638 - 0.226i)T + (-0.852 + 0.522i)T^{2} \)
17 \( 1 + (-0.760 - 0.649i)T^{2} \)
19 \( 1 + (0.532 - 0.105i)T + (0.923 - 0.382i)T^{2} \)
23 \( 1 + (-0.233 + 0.972i)T^{2} \)
29 \( 1 + (0.891 + 0.453i)T^{2} \)
31 \( 1 + (1.04 + 0.384i)T + (0.760 + 0.649i)T^{2} \)
37 \( 1 + (1.41 + 0.398i)T + (0.852 + 0.522i)T^{2} \)
41 \( 1 + (0.891 - 0.453i)T^{2} \)
43 \( 1 + (0.568 + 0.614i)T + (-0.0784 + 0.996i)T^{2} \)
47 \( 1 + (-0.891 - 0.453i)T^{2} \)
53 \( 1 + (-0.453 + 0.891i)T^{2} \)
59 \( 1 + (-0.987 - 0.156i)T^{2} \)
61 \( 1 + (-0.0245 - 0.311i)T + (-0.987 + 0.156i)T^{2} \)
67 \( 1 + (0.882 - 0.211i)T + (0.891 - 0.453i)T^{2} \)
71 \( 1 + (0.996 + 0.0784i)T^{2} \)
73 \( 1 + (1.80 + 0.509i)T + (0.852 + 0.522i)T^{2} \)
79 \( 1 + (-1.89 - 0.149i)T + (0.987 + 0.156i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (-0.382 + 0.923i)T^{2} \)
97 \( 1 + (-0.203 - 1.28i)T + (-0.951 + 0.309i)T^{2} \)
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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

−8.985431289425639260515993516994, −8.472755049426308043936317649970, −7.76372680107908902304071273940, −7.19188248775292772746804461106, −6.28163139288104539273104604684, −5.29095654111979024046694994489, −4.43033465933884965227444882630, −3.63076366736344203870915725420, −2.33141637888758020791150789821, −1.65401479215701644256961147766, 1.40892451801250106023374922882, 2.03267040591838189891514802880, 3.29814531558830906793715219718, 4.62055392410690564816117878167, 5.21014374670609624246992711064, 5.92578881075619208199182037529, 6.98013972985576700349941044547, 7.56017347701864781685291030333, 8.415804075826640071339754730540, 9.090122301865643193105650846119

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