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Properties

Label 2-39e2-169.83-c0-0-0
Degree $2$
Conductor $1521$
Sign $0.284 + 0.958i$
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  + (0.239 − 0.970i)4-s + (−0.110 + 0.0495i)7-s − i·13-s + (−0.885 − 0.464i)16-s + (0.872 − 0.872i)19-s + (0.822 + 0.568i)25-s + (0.0217 + 0.118i)28-s + (−1.68 − 0.308i)31-s + (0.328 − 1.79i)37-s + (1.53 + 1.06i)43-s + (−0.653 + 0.737i)49-s + (−0.970 − 0.239i)52-s + (−0.169 − 0.447i)61-s + (−0.663 + 0.748i)64-s + (−0.186 + 0.308i)67-s + ⋯
L(s)  = 1  + (0.239 − 0.970i)4-s + (−0.110 + 0.0495i)7-s − i·13-s + (−0.885 − 0.464i)16-s + (0.872 − 0.872i)19-s + (0.822 + 0.568i)25-s + (0.0217 + 0.118i)28-s + (−1.68 − 0.308i)31-s + (0.328 − 1.79i)37-s + (1.53 + 1.06i)43-s + (−0.653 + 0.737i)49-s + (−0.970 − 0.239i)52-s + (−0.169 − 0.447i)61-s + (−0.663 + 0.748i)64-s + (−0.186 + 0.308i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.284 + 0.958i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (928, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.284 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.116161943\)
\(L(\frac12)\) \(\approx\) \(1.116161943\)
\(L(1)\) 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 \( 1 + (-0.239 + 0.970i)T^{2} \)
5 \( 1 + (-0.822 - 0.568i)T^{2} \)
7 \( 1 + (0.110 - 0.0495i)T + (0.663 - 0.748i)T^{2} \)
11 \( 1 + (-0.239 - 0.970i)T^{2} \)
17 \( 1 + (0.748 + 0.663i)T^{2} \)
19 \( 1 + (-0.872 + 0.872i)T - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.970 - 0.239i)T^{2} \)
31 \( 1 + (1.68 + 0.308i)T + (0.935 + 0.354i)T^{2} \)
37 \( 1 + (-0.328 + 1.79i)T + (-0.935 - 0.354i)T^{2} \)
41 \( 1 + (-0.992 - 0.120i)T^{2} \)
43 \( 1 + (-1.53 - 1.06i)T + (0.354 + 0.935i)T^{2} \)
47 \( 1 + (0.464 + 0.885i)T^{2} \)
53 \( 1 + (-0.748 - 0.663i)T^{2} \)
59 \( 1 + (0.822 + 0.568i)T^{2} \)
61 \( 1 + (0.169 + 0.447i)T + (-0.748 + 0.663i)T^{2} \)
67 \( 1 + (0.186 - 0.308i)T + (-0.464 - 0.885i)T^{2} \)
71 \( 1 + (-0.992 - 0.120i)T^{2} \)
73 \( 1 + (-0.366 + 0.468i)T + (-0.239 - 0.970i)T^{2} \)
79 \( 1 + (1.28 - 0.317i)T + (0.885 - 0.464i)T^{2} \)
83 \( 1 + (0.992 - 0.120i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-1.35 + 0.420i)T + (0.822 - 0.568i)T^{2} \)
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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

−9.360044745222963883166314934827, −9.064130680441690691009530863456, −7.67376404865480764423959930792, −7.16937956630856931966953263265, −6.05635313846568041720866902058, −5.48444124955071088794758111780, −4.66938758932395390655109700448, −3.34958144391925442754992654812, −2.32747429692897327869195696195, −0.944443635658762562840125171169, 1.75400301132838626162753376692, 2.95242592994860468110692655560, 3.79753347353277574956374230350, 4.66687049107100476454777916922, 5.81199619193577708890939584076, 6.81472959696374416789525028911, 7.34446840246419510222270846676, 8.249119799944342887710908355995, 8.930431738377053565979475431528, 9.712841560206854103728267711129

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