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Properties

Label 2-3459-3459.266-c0-0-0
Degree $2$
Conductor $3459$
Sign $-0.862 - 0.505i$
Analytic cond. $1.72626$
Root an. cond. $1.31387$
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  + (0.582 − 0.812i)3-s + (0.130 + 0.991i)4-s + (−1.21 + 1.10i)7-s + (−0.321 − 0.946i)9-s + (0.881 + 0.471i)12-s + (−1.52 − 1.25i)13-s + (−0.965 + 0.258i)16-s + (−1.36 + 1.08i)19-s + (0.187 + 1.63i)21-s + (−0.162 + 0.986i)25-s + (−0.956 − 0.290i)27-s + (−1.25 − 1.06i)28-s + (0.0699 − 0.607i)31-s + (0.896 − 0.442i)36-s + (0.346 + 0.968i)37-s + ⋯
L(s)  = 1  + (0.582 − 0.812i)3-s + (0.130 + 0.991i)4-s + (−1.21 + 1.10i)7-s + (−0.321 − 0.946i)9-s + (0.881 + 0.471i)12-s + (−1.52 − 1.25i)13-s + (−0.965 + 0.258i)16-s + (−1.36 + 1.08i)19-s + (0.187 + 1.63i)21-s + (−0.162 + 0.986i)25-s + (−0.956 − 0.290i)27-s + (−1.25 − 1.06i)28-s + (0.0699 − 0.607i)31-s + (0.896 − 0.442i)36-s + (0.346 + 0.968i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3459\)    =    \(3 \cdot 1153\)
Sign: $-0.862 - 0.505i$
Analytic conductor: \(1.72626\)
Root analytic conductor: \(1.31387\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3459} (266, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3459,\ (\ :0),\ -0.862 - 0.505i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3879572453\)
\(L(\frac12)\) \(\approx\) \(0.3879572453\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.582 + 0.812i)T \)
1153 \( 1 + (-0.0980 + 0.995i)T \)
good2 \( 1 + (-0.130 - 0.991i)T^{2} \)
5 \( 1 + (0.162 - 0.986i)T^{2} \)
7 \( 1 + (1.21 - 1.10i)T + (0.0980 - 0.995i)T^{2} \)
11 \( 1 + (0.997 - 0.0654i)T^{2} \)
13 \( 1 + (1.52 + 1.25i)T + (0.195 + 0.980i)T^{2} \)
17 \( 1 + (-0.729 - 0.683i)T^{2} \)
19 \( 1 + (1.36 - 1.08i)T + (0.227 - 0.973i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.321 + 0.946i)T^{2} \)
31 \( 1 + (-0.0699 + 0.607i)T + (-0.973 - 0.227i)T^{2} \)
37 \( 1 + (-0.346 - 0.968i)T + (-0.773 + 0.634i)T^{2} \)
41 \( 1 + (-0.946 - 0.321i)T^{2} \)
43 \( 1 + (0.322 - 1.06i)T + (-0.831 - 0.555i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.634 - 0.773i)T^{2} \)
61 \( 1 + (-0.878 - 0.415i)T + (0.634 + 0.773i)T^{2} \)
67 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (-0.290 - 0.956i)T^{2} \)
73 \( 1 + (-0.540 - 0.0983i)T + (0.935 + 0.352i)T^{2} \)
79 \( 1 + (1.44 + 0.864i)T + (0.471 + 0.881i)T^{2} \)
83 \( 1 + (0.812 - 0.582i)T^{2} \)
89 \( 1 + (-0.0654 - 0.997i)T^{2} \)
97 \( 1 + (0.162 + 0.282i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.898227568209050351846696452974, −8.233928229339933792197596136192, −7.69981238393191795978566460448, −6.96483050300026690428220979655, −6.21930491217935679335925696645, −5.56542460855267540319241686159, −4.25734890032703745746876510688, −3.15636292891074536752369794961, −2.84356728593812807038417955513, −2.00049980571108004663121157683, 0.18354460161192878671344806323, 2.06393045911768378603625043265, 2.74927311069494886403446709483, 4.02789913866688982315808135151, 4.45094824160521695650800244928, 5.24355418749942995922417282990, 6.44523647186173418782108819841, 6.83578149064424841635872514069, 7.58590465996649037149701247920, 8.865734968051007838340547379426

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