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Properties

Label 2-39e2-13.6-c0-0-3
Degree $2$
Conductor $1521$
Sign $0.522 + 0.852i$
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  + (0.866 − 0.5i)4-s + (0.366 − 1.36i)7-s + (0.499 − 0.866i)16-s + (−1.36 − 0.366i)19-s + i·25-s + (−0.366 − 1.36i)28-s + (−1 + i)31-s + (1.36 − 0.366i)37-s + (−0.866 − 0.5i)49-s − 0.999i·64-s + (0.366 + 1.36i)67-s + (1 + i)73-s + (−1.36 + 0.366i)76-s + (1.36 + 0.366i)97-s + (0.5 + 0.866i)100-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)4-s + (0.366 − 1.36i)7-s + (0.499 − 0.866i)16-s + (−1.36 − 0.366i)19-s + i·25-s + (−0.366 − 1.36i)28-s + (−1 + i)31-s + (1.36 − 0.366i)37-s + (−0.866 − 0.5i)49-s − 0.999i·64-s + (0.366 + 1.36i)67-s + (1 + i)73-s + (−1.36 + 0.366i)76-s + (1.36 + 0.366i)97-s + (0.5 + 0.866i)100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.353289440\)
\(L(\frac12)\) \(\approx\) \(1.353289440\)
\(L(1)\) 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 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
5 \( 1 - iT^{2} \)
7 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (1 - i)T - iT^{2} \)
37 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.720416039731362216609652004643, −8.717811328262210824168139723508, −7.68700983322769198851565354339, −7.12222443154782787308703068223, −6.44100172084808297158730745855, −5.45057657107226872241429111748, −4.48212007270362702465734413992, −3.54521202356348720887550044278, −2.26939017078099746105840395363, −1.15383377752205313082012180048, 1.98472107510920916283955765044, 2.53367718792264570517182160135, 3.73748892601373558384545000089, 4.81268408536586929100584542989, 6.01438443642342500997404719769, 6.30712022747573352037572162635, 7.52207748812941015621387774651, 8.208760516460677322969968734086, 8.807394424942712704815718900921, 9.745951587489432552726436005742

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