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Properties

Label 2-39e2-169.161-c0-0-0
Degree $2$
Conductor $1521$
Sign $0.690 - 0.723i$
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  + (0.992 + 0.120i)4-s + (−0.328 + 1.79i)7-s − i·13-s + (0.970 + 0.239i)16-s + (0.0853 − 0.0853i)19-s + (0.464 + 0.885i)25-s + (−0.542 + 1.74i)28-s + (−1.17 + 0.366i)31-s + (−0.107 − 0.344i)37-s + (0.764 + 1.45i)43-s + (−2.17 − 0.824i)49-s + (0.120 − 0.992i)52-s + (1.12 − 1.63i)61-s + (0.935 + 0.354i)64-s + (−0.468 − 0.366i)67-s + ⋯
L(s)  = 1  + (0.992 + 0.120i)4-s + (−0.328 + 1.79i)7-s − i·13-s + (0.970 + 0.239i)16-s + (0.0853 − 0.0853i)19-s + (0.464 + 0.885i)25-s + (−0.542 + 1.74i)28-s + (−1.17 + 0.366i)31-s + (−0.107 − 0.344i)37-s + (0.764 + 1.45i)43-s + (−2.17 − 0.824i)49-s + (0.120 − 0.992i)52-s + (1.12 − 1.63i)61-s + (0.935 + 0.354i)64-s + (−0.468 − 0.366i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.690 - 0.723i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1513, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.690 - 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.339114184\)
\(L(\frac12)\) \(\approx\) \(1.339114184\)
\(L(1)\) 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 \( 1 + (-0.992 - 0.120i)T^{2} \)
5 \( 1 + (-0.464 - 0.885i)T^{2} \)
7 \( 1 + (0.328 - 1.79i)T + (-0.935 - 0.354i)T^{2} \)
11 \( 1 + (-0.992 + 0.120i)T^{2} \)
17 \( 1 + (0.354 - 0.935i)T^{2} \)
19 \( 1 + (-0.0853 + 0.0853i)T - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.120 - 0.992i)T^{2} \)
31 \( 1 + (1.17 - 0.366i)T + (0.822 - 0.568i)T^{2} \)
37 \( 1 + (0.107 + 0.344i)T + (-0.822 + 0.568i)T^{2} \)
41 \( 1 + (0.663 + 0.748i)T^{2} \)
43 \( 1 + (-0.764 - 1.45i)T + (-0.568 + 0.822i)T^{2} \)
47 \( 1 + (-0.239 - 0.970i)T^{2} \)
53 \( 1 + (-0.354 + 0.935i)T^{2} \)
59 \( 1 + (0.464 + 0.885i)T^{2} \)
61 \( 1 + (-1.12 + 1.63i)T + (-0.354 - 0.935i)T^{2} \)
67 \( 1 + (0.468 + 0.366i)T + (0.239 + 0.970i)T^{2} \)
71 \( 1 + (0.663 + 0.748i)T^{2} \)
73 \( 1 + (0.0624 + 1.03i)T + (-0.992 + 0.120i)T^{2} \)
79 \( 1 + (0.225 + 1.85i)T + (-0.970 + 0.239i)T^{2} \)
83 \( 1 + (-0.663 + 0.748i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (1.21 - 0.731i)T + (0.464 - 0.885i)T^{2} \)
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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

−9.651492856190573185282322353125, −8.986457759875551171925435281436, −8.145203202470285062563988302954, −7.38358925705073005546458715816, −6.39324249007940974377335930535, −5.73882381053105204727723410381, −5.09868205705459501983288419583, −3.38846305101670161678014753649, −2.78109598653164537053829581921, −1.78909631587724474849408480976, 1.15923454046671906569672505096, 2.40010653458485275626929272098, 3.65872567950834281367438105957, 4.28799531832515262289234680130, 5.56674273756453533142231948568, 6.62600923522148655963863228512, 7.04619848066126142886613909129, 7.66654724352175536476748377838, 8.743930409525645367823659291272, 9.832308909800622727647105352122

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