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Properties

Label 2-3459-3459.707-c0-0-0
Degree $2$
Conductor $3459$
Sign $-0.725 - 0.688i$
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  + (−0.986 − 0.162i)3-s + (−0.130 + 0.991i)4-s + (0.142 − 0.397i)7-s + (0.946 + 0.321i)9-s + (0.290 − 0.956i)12-s + (−1.61 − 0.159i)13-s + (−0.965 − 0.258i)16-s + (1.19 + 0.341i)19-s + (−0.205 + 0.369i)21-s + (−0.582 + 0.812i)25-s + (−0.881 − 0.471i)27-s + (0.375 + 0.192i)28-s + (0.740 + 1.33i)31-s + (−0.442 + 0.896i)36-s + (−0.293 + 0.0143i)37-s + ⋯
L(s)  = 1  + (−0.986 − 0.162i)3-s + (−0.130 + 0.991i)4-s + (0.142 − 0.397i)7-s + (0.946 + 0.321i)9-s + (0.290 − 0.956i)12-s + (−1.61 − 0.159i)13-s + (−0.965 − 0.258i)16-s + (1.19 + 0.341i)19-s + (−0.205 + 0.369i)21-s + (−0.582 + 0.812i)25-s + (−0.881 − 0.471i)27-s + (0.375 + 0.192i)28-s + (0.740 + 1.33i)31-s + (−0.442 + 0.896i)36-s + (−0.293 + 0.0143i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3459\)    =    \(3 \cdot 1153\)
Sign: $-0.725 - 0.688i$
Analytic conductor: \(1.72626\)
Root analytic conductor: \(1.31387\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3459} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3459,\ (\ :0),\ -0.725 - 0.688i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4741126507\)
\(L(\frac12)\) \(\approx\) \(0.4741126507\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.986 + 0.162i)T \)
1153 \( 1 + (0.773 + 0.634i)T \)
good2 \( 1 + (0.130 - 0.991i)T^{2} \)
5 \( 1 + (0.582 - 0.812i)T^{2} \)
7 \( 1 + (-0.142 + 0.397i)T + (-0.773 - 0.634i)T^{2} \)
11 \( 1 + (0.0654 - 0.997i)T^{2} \)
13 \( 1 + (1.61 + 0.159i)T + (0.980 + 0.195i)T^{2} \)
17 \( 1 + (-0.0327 + 0.999i)T^{2} \)
19 \( 1 + (-1.19 - 0.341i)T + (0.849 + 0.528i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.946 - 0.321i)T^{2} \)
31 \( 1 + (-0.740 - 1.33i)T + (-0.528 + 0.849i)T^{2} \)
37 \( 1 + (0.293 - 0.0143i)T + (0.995 - 0.0980i)T^{2} \)
41 \( 1 + (-0.321 - 0.946i)T^{2} \)
43 \( 1 + (0.783 - 1.46i)T + (-0.555 - 0.831i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.0980 - 0.995i)T^{2} \)
61 \( 1 + (1.16 + 1.05i)T + (0.0980 + 0.995i)T^{2} \)
67 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
71 \( 1 + (0.471 + 0.881i)T^{2} \)
73 \( 1 + (-0.123 - 0.192i)T + (-0.412 + 0.910i)T^{2} \)
79 \( 1 + (1.62 - 0.241i)T + (0.956 - 0.290i)T^{2} \)
83 \( 1 + (-0.162 - 0.986i)T^{2} \)
89 \( 1 + (0.997 + 0.0654i)T^{2} \)
97 \( 1 + (0.582 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.133691614265765216983303148303, −7.972275168969157261070717737908, −7.50498209025795754712694858523, −7.03198101739722521011671927288, −6.10751412666606213108177011159, −5.01079266859009589955114902252, −4.72825007181927875962437455948, −3.62416753324827588834145289442, −2.74398842038952326676646259259, −1.41951650829817441554722588661, 0.32537034068287857256246594901, 1.68701542928299184163890398971, 2.69259867402710330461069823407, 4.22888555166242798052515365067, 4.78887978255614219041720736704, 5.54651659636114943503020369175, 5.96556804650892261860865729862, 6.99743801110641805998219360469, 7.43821164195016945429634817326, 8.655303340451021486457931444646

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