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Properties

Label 2-57e2-171.103-c0-0-0
Degree $2$
Conductor $3249$
Sign $-0.377 - 0.925i$
Analytic cond. $1.62146$
Root an. cond. $1.27336$
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  + i·2-s + (0.866 + 0.5i)3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + i·8-s + (0.499 + 0.866i)9-s + (0.866 − 0.5i)10-s + (0.5 + 0.866i)11-s + i·13-s + (0.866 − 0.5i)14-s − 0.999i·15-s − 16-s + (−0.866 + 0.499i)18-s − 0.999i·21-s + (−0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + i·2-s + (0.866 + 0.5i)3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + i·8-s + (0.499 + 0.866i)9-s + (0.866 − 0.5i)10-s + (0.5 + 0.866i)11-s + i·13-s + (0.866 − 0.5i)14-s − 0.999i·15-s − 16-s + (−0.866 + 0.499i)18-s − 0.999i·21-s + (−0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3249\)    =    \(3^{2} \cdot 19^{2}\)
Sign: $-0.377 - 0.925i$
Analytic conductor: \(1.62146\)
Root analytic conductor: \(1.27336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3249} (3181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3249,\ (\ :0),\ -0.377 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.692151276\)
\(L(\frac12)\) \(\approx\) \(1.692151276\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 \)
good2 \( 1 - iT - T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−8.977440747385035824082096061633, −8.136805135300413564064923616531, −7.68132970987373296332605561326, −6.88544946830315770653054332533, −6.36703637482567675088751227017, −5.07139296908961610999724303546, −4.40206185632107883636033408269, −3.95280525624727366517558353730, −2.66876054969257525298115112071, −1.55055409765672252724664728835, 0.970849694995781042071026792199, 2.36492469892885217930073532316, 2.84619274314538837040087896126, 3.44203264323633921580717397823, 4.18884055683482022185798452812, 5.90449159526958471522330074513, 6.34849943238090450427996119859, 7.18055456737483649394202925331, 7.960585911828243385162897426292, 8.599805546333105722036475644106

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