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Properties

Label 2-3311-3311.937-c0-0-1
Degree $2$
Conductor $3311$
Sign $0.706 - 0.707i$
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  + (0.984 − 0.369i)2-s + (0.0793 − 0.0693i)4-s + (0.913 − 0.406i)7-s + (−0.445 + 0.828i)8-s + (−0.575 + 0.817i)9-s + (−0.134 + 0.990i)11-s + (0.748 − 0.737i)14-s + (−0.146 + 1.08i)16-s + (−0.264 + 1.01i)18-s + (0.233 + 1.02i)22-s + (−0.494 + 0.458i)23-s + (0.842 + 0.538i)25-s + (0.0442 − 0.0956i)28-s + (−0.0265 − 0.0137i)29-s + (0.0467 + 0.204i)32-s + ⋯
L(s)  = 1  + (0.984 − 0.369i)2-s + (0.0793 − 0.0693i)4-s + (0.913 − 0.406i)7-s + (−0.445 + 0.828i)8-s + (−0.575 + 0.817i)9-s + (−0.134 + 0.990i)11-s + (0.748 − 0.737i)14-s + (−0.146 + 1.08i)16-s + (−0.264 + 1.01i)18-s + (0.233 + 1.02i)22-s + (−0.494 + 0.458i)23-s + (0.842 + 0.538i)25-s + (0.0442 − 0.0956i)28-s + (−0.0265 − 0.0137i)29-s + (0.0467 + 0.204i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.706 - 0.707i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (937, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 0.706 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.888673363\)
\(L(\frac12)\) \(\approx\) \(1.888673363\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.913 + 0.406i)T \)
11 \( 1 + (0.134 - 0.990i)T \)
43 \( 1 + (0.420 - 0.907i)T \)
good2 \( 1 + (-0.984 + 0.369i)T + (0.753 - 0.657i)T^{2} \)
3 \( 1 + (0.575 - 0.817i)T^{2} \)
5 \( 1 + (-0.842 - 0.538i)T^{2} \)
13 \( 1 + (0.712 + 0.701i)T^{2} \)
17 \( 1 + (0.251 + 0.967i)T^{2} \)
19 \( 1 + (0.280 + 0.959i)T^{2} \)
23 \( 1 + (0.494 - 0.458i)T + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (0.0265 + 0.0137i)T + (0.575 + 0.817i)T^{2} \)
31 \( 1 + (-0.193 - 0.981i)T^{2} \)
37 \( 1 + (0.355 + 0.0373i)T + (0.978 + 0.207i)T^{2} \)
41 \( 1 + (0.858 + 0.512i)T^{2} \)
47 \( 1 + (0.691 - 0.722i)T^{2} \)
53 \( 1 + (-0.260 - 0.0636i)T + (0.887 + 0.460i)T^{2} \)
59 \( 1 + (0.550 - 0.834i)T^{2} \)
61 \( 1 + (-0.193 + 0.981i)T^{2} \)
67 \( 1 + (-1.90 + 0.588i)T + (0.826 - 0.563i)T^{2} \)
71 \( 1 + (-0.377 - 0.174i)T + (0.646 + 0.762i)T^{2} \)
73 \( 1 + (0.163 + 0.986i)T^{2} \)
79 \( 1 + (-1.34 + 1.21i)T + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (-0.946 + 0.323i)T^{2} \)
89 \( 1 + (0.955 + 0.294i)T^{2} \)
97 \( 1 + (-0.983 - 0.178i)T^{2} \)
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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

−8.781664029831844950857364865859, −8.056474790482451968922750627197, −7.56090383941357712993934745821, −6.56323109421160491439356150727, −5.42502784663616859896694024067, −5.02341370218048041587527855077, −4.37580514158290387913757797495, −3.50132687442741130147660142927, −2.48467938117664238568293956750, −1.71452773331424682148109835621, 0.839703680472295738208274461584, 2.43780363084334631908268509264, 3.39186577530387948096598644283, 4.11179627851984312254495464503, 5.08315115007678049547857687944, 5.55437742147782260320824333943, 6.30746758609111166053040885048, 6.89917273674993022771805998690, 8.101008996689287334029616194163, 8.621672662287154628047058047833

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