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Properties

Label 2-3311-3311.1525-c0-0-1
Degree $2$
Conductor $3311$
Sign $-0.136 - 0.990i$
Analytic cond. $1.65240$
Root an. cond. $1.28545$
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  + (−0.0709 − 1.58i)2-s + (−1.49 + 0.134i)4-s + (0.669 − 0.743i)7-s + (0.106 + 0.788i)8-s + (−0.992 + 0.119i)9-s + (−0.983 + 0.178i)11-s + (−1.22 − 1.00i)14-s + (−0.239 + 0.0435i)16-s + (0.259 + 1.56i)18-s + (0.352 + 1.54i)22-s + (−1.85 − 0.572i)23-s + (0.447 − 0.894i)25-s + (−0.901 + 1.20i)28-s + (−0.385 − 0.0231i)29-s + (0.262 + 1.15i)32-s + ⋯
L(s)  = 1  + (−0.0709 − 1.58i)2-s + (−1.49 + 0.134i)4-s + (0.669 − 0.743i)7-s + (0.106 + 0.788i)8-s + (−0.992 + 0.119i)9-s + (−0.983 + 0.178i)11-s + (−1.22 − 1.00i)14-s + (−0.239 + 0.0435i)16-s + (0.259 + 1.56i)18-s + (0.352 + 1.54i)22-s + (−1.85 − 0.572i)23-s + (0.447 − 0.894i)25-s + (−0.901 + 1.20i)28-s + (−0.385 − 0.0231i)29-s + (0.262 + 1.15i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.136 - 0.990i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1525, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.136 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4256434737\)
\(L(\frac12)\) \(\approx\) \(0.4256434737\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.669 + 0.743i)T \)
11 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (-0.599 + 0.800i)T \)
good2 \( 1 + (0.0709 + 1.58i)T + (-0.995 + 0.0896i)T^{2} \)
3 \( 1 + (0.992 - 0.119i)T^{2} \)
5 \( 1 + (-0.447 + 0.894i)T^{2} \)
13 \( 1 + (-0.772 + 0.635i)T^{2} \)
17 \( 1 + (-0.163 + 0.986i)T^{2} \)
19 \( 1 + (-0.525 + 0.850i)T^{2} \)
23 \( 1 + (1.85 + 0.572i)T + (0.826 + 0.563i)T^{2} \)
29 \( 1 + (0.385 + 0.0231i)T + (0.992 + 0.119i)T^{2} \)
31 \( 1 + (-0.575 - 0.817i)T^{2} \)
37 \( 1 + (0.300 - 1.41i)T + (-0.913 - 0.406i)T^{2} \)
41 \( 1 + (0.753 - 0.657i)T^{2} \)
47 \( 1 + (-0.473 - 0.880i)T^{2} \)
53 \( 1 + (1.96 + 0.0588i)T + (0.998 + 0.0598i)T^{2} \)
59 \( 1 + (0.963 + 0.266i)T^{2} \)
61 \( 1 + (-0.575 + 0.817i)T^{2} \)
67 \( 1 + (1.04 + 0.969i)T + (0.0747 + 0.997i)T^{2} \)
71 \( 1 + (-0.650 - 0.487i)T + (0.280 + 0.959i)T^{2} \)
73 \( 1 + (0.842 + 0.538i)T^{2} \)
79 \( 1 + (1.59 + 0.167i)T + (0.978 + 0.207i)T^{2} \)
83 \( 1 + (0.420 + 0.907i)T^{2} \)
89 \( 1 + (-0.733 + 0.680i)T^{2} \)
97 \( 1 + (0.691 + 0.722i)T^{2} \)
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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

−8.199097945110116907073035024600, −8.039972680689060375670112335314, −6.80624956692313038409411134320, −5.81046613164821454586266306151, −4.82569371705268518897797021073, −4.25930466684387103998379326203, −3.27095332746010225553622264106, −2.48740061158363668992614638464, −1.68307682915165505216562173671, −0.24105590630640827800994882164, 2.01408706815012205798217497046, 3.04445050628373117369683322072, 4.30769466885702961316494346708, 5.23969859656083386459439864568, 5.70069767700485964925193954137, 6.14240969414724435793197118804, 7.28712213531929055605926761065, 7.892943889801169387367377000070, 8.312948143454298449162275382382, 9.046138728980964348542574005126

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