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Properties

Label 2-39e2-169.60-c0-0-0
Degree $2$
Conductor $1521$
Sign $0.337 - 0.941i$
Analytic cond. $0.759077$
Root an. cond. $0.871250$
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  + (−0.663 + 0.748i)4-s + (0.107 − 0.344i)7-s + i·13-s + (−0.120 − 0.992i)16-s + (1.28 + 1.28i)19-s + (−0.239 + 0.970i)25-s + (0.186 + 0.308i)28-s + (0.103 + 0.0624i)31-s + (−0.307 + 0.509i)37-s + (−0.222 + 0.902i)43-s + (0.716 + 0.494i)49-s + (−0.748 − 0.663i)52-s + (−1.17 − 0.616i)61-s + (0.822 + 0.568i)64-s + (−1.03 − 0.0624i)67-s + ⋯
L(s)  = 1  + (−0.663 + 0.748i)4-s + (0.107 − 0.344i)7-s + i·13-s + (−0.120 − 0.992i)16-s + (1.28 + 1.28i)19-s + (−0.239 + 0.970i)25-s + (0.186 + 0.308i)28-s + (0.103 + 0.0624i)31-s + (−0.307 + 0.509i)37-s + (−0.222 + 0.902i)43-s + (0.716 + 0.494i)49-s + (−0.748 − 0.663i)52-s + (−1.17 − 0.616i)61-s + (0.822 + 0.568i)64-s + (−1.03 − 0.0624i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.337 - 0.941i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.337 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9102903334\)
\(L(\frac12)\) \(\approx\) \(0.9102903334\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 - iT \)
good2 \( 1 + (0.663 - 0.748i)T^{2} \)
5 \( 1 + (0.239 - 0.970i)T^{2} \)
7 \( 1 + (-0.107 + 0.344i)T + (-0.822 - 0.568i)T^{2} \)
11 \( 1 + (0.663 + 0.748i)T^{2} \)
17 \( 1 + (-0.568 + 0.822i)T^{2} \)
19 \( 1 + (-1.28 - 1.28i)T + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.748 - 0.663i)T^{2} \)
31 \( 1 + (-0.103 - 0.0624i)T + (0.464 + 0.885i)T^{2} \)
37 \( 1 + (0.307 - 0.509i)T + (-0.464 - 0.885i)T^{2} \)
41 \( 1 + (-0.935 - 0.354i)T^{2} \)
43 \( 1 + (0.222 - 0.902i)T + (-0.885 - 0.464i)T^{2} \)
47 \( 1 + (-0.992 - 0.120i)T^{2} \)
53 \( 1 + (0.568 - 0.822i)T^{2} \)
59 \( 1 + (-0.239 + 0.970i)T^{2} \)
61 \( 1 + (1.17 + 0.616i)T + (0.568 + 0.822i)T^{2} \)
67 \( 1 + (1.03 + 0.0624i)T + (0.992 + 0.120i)T^{2} \)
71 \( 1 + (-0.935 - 0.354i)T^{2} \)
73 \( 1 + (-1.43 - 0.646i)T + (0.663 + 0.748i)T^{2} \)
79 \( 1 + (-1.23 + 1.09i)T + (0.120 - 0.992i)T^{2} \)
83 \( 1 + (0.935 - 0.354i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.872 - 1.11i)T + (-0.239 - 0.970i)T^{2} \)
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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

−9.551574132949492246386096349194, −9.170610047849371106811285664529, −8.085489956139273103502800863814, −7.60442033882115999419853831820, −6.73075810840570251069757215080, −5.60560378143668655054816078045, −4.69741643320965982034170295102, −3.85741770871148597086293358314, −3.08660161286673216358731466821, −1.52400434727365606286538217951, 0.817351946736378382847572659876, 2.34374337848730556997047808468, 3.51032816890199390163874865968, 4.68778148624933660337260889428, 5.33712448446303012817129064588, 6.04538528905465612968192028720, 7.09242708939046193846385463635, 8.021543495304060901207618120141, 8.851422051902496369051698506984, 9.451537769640878138840603303676

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