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Properties

Label 2-3311-3311.426-c0-0-0
Degree $2$
Conductor $3311$
Sign $-0.902 + 0.430i$
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  + (−0.0620 + 1.38i)2-s + (−0.906 − 0.0815i)4-s + (−0.309 + 0.951i)7-s + (−0.0166 + 0.122i)8-s + (0.393 + 0.919i)9-s + (0.0448 + 0.998i)11-s + (−1.29 − 0.485i)14-s + (−1.06 − 0.193i)16-s + (−1.29 + 0.485i)18-s − 1.38·22-s + (0.0597 − 0.261i)23-s + (0.550 − 0.834i)25-s + (0.357 − 0.837i)28-s + (0.433 + 0.656i)29-s + (0.305 − 1.33i)32-s + ⋯
L(s)  = 1  + (−0.0620 + 1.38i)2-s + (−0.906 − 0.0815i)4-s + (−0.309 + 0.951i)7-s + (−0.0166 + 0.122i)8-s + (0.393 + 0.919i)9-s + (0.0448 + 0.998i)11-s + (−1.29 − 0.485i)14-s + (−1.06 − 0.193i)16-s + (−1.29 + 0.485i)18-s − 1.38·22-s + (0.0597 − 0.261i)23-s + (0.550 − 0.834i)25-s + (0.357 − 0.837i)28-s + (0.433 + 0.656i)29-s + (0.305 − 1.33i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.902 + 0.430i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (426, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.902 + 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.070657087\)
\(L(\frac12)\) \(\approx\) \(1.070657087\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.0448 - 0.998i)T \)
43 \( 1 + (-0.753 - 0.657i)T \)
good2 \( 1 + (0.0620 - 1.38i)T + (-0.995 - 0.0896i)T^{2} \)
3 \( 1 + (-0.393 - 0.919i)T^{2} \)
5 \( 1 + (-0.550 + 0.834i)T^{2} \)
13 \( 1 + (0.936 - 0.351i)T^{2} \)
17 \( 1 + (0.936 + 0.351i)T^{2} \)
19 \( 1 + (-0.473 + 0.880i)T^{2} \)
23 \( 1 + (-0.0597 + 0.261i)T + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (-0.433 - 0.656i)T + (-0.393 + 0.919i)T^{2} \)
31 \( 1 + (0.995 + 0.0896i)T^{2} \)
37 \( 1 + (1.88 + 0.612i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.753 + 0.657i)T^{2} \)
47 \( 1 + (-0.473 + 0.880i)T^{2} \)
53 \( 1 + (-0.449 + 0.834i)T + (-0.550 - 0.834i)T^{2} \)
59 \( 1 + (0.963 - 0.266i)T^{2} \)
61 \( 1 + (0.995 - 0.0896i)T^{2} \)
67 \( 1 + (-0.0199 - 0.0874i)T + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (-1.08 - 0.462i)T + (0.691 + 0.722i)T^{2} \)
73 \( 1 + (0.0448 + 0.998i)T^{2} \)
79 \( 1 + (-0.773 - 1.06i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.995 + 0.0896i)T^{2} \)
89 \( 1 + (-0.222 + 0.974i)T^{2} \)
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

−8.920603608381530366985227055172, −8.373301655540314866846781434409, −7.67411349592384227464838915788, −6.88509860097020651452752707727, −6.50148030144877806396494477736, −5.37571327669348492055008135823, −5.08214096594119935952663754374, −4.15803399904835258610124566159, −2.70432789684219898071900626585, −1.93282181699900988152627389285, 0.66798856675188381838487741232, 1.51770508826312807358448811450, 2.86065597631804834946353261628, 3.56728782285526806707685953406, 4.01100357292043284100566552067, 5.09826160853218479957816963167, 6.27858389511335609204591617184, 6.83646609146071569180053933551, 7.65862622431924240808736602115, 8.801341497718922981307624590471

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