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Properties

Label 2-3332-3332.2923-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.138 + 0.990i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (−0.0747 − 0.997i)2-s + (−0.563 − 0.173i)3-s + (−0.988 + 0.149i)4-s + (−0.131 + 0.574i)6-s + (−0.974 + 0.222i)7-s + (0.222 + 0.974i)8-s + (−0.539 − 0.367i)9-s + (−1.64 + 1.12i)11-s + (0.582 + 0.0878i)12-s + (0.134 + 0.0648i)13-s + (0.294 + 0.955i)14-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−0.326 + 0.565i)18-s + (0.587 + 0.0440i)21-s + (1.24 + 1.55i)22-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.563 − 0.173i)3-s + (−0.988 + 0.149i)4-s + (−0.131 + 0.574i)6-s + (−0.974 + 0.222i)7-s + (0.222 + 0.974i)8-s + (−0.539 − 0.367i)9-s + (−1.64 + 1.12i)11-s + (0.582 + 0.0878i)12-s + (0.134 + 0.0648i)13-s + (0.294 + 0.955i)14-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−0.326 + 0.565i)18-s + (0.587 + 0.0440i)21-s + (1.24 + 1.55i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.138 + 0.990i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2923, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5392196690\)
\(L(\frac12)\) \(\approx\) \(0.5392196690\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0747 + 0.997i)T \)
7 \( 1 + (0.974 - 0.222i)T \)
17 \( 1 + (-0.365 - 0.930i)T \)
good3 \( 1 + (0.563 + 0.173i)T + (0.826 + 0.563i)T^{2} \)
5 \( 1 + (-0.0747 + 0.997i)T^{2} \)
11 \( 1 + (1.64 - 1.12i)T + (0.365 - 0.930i)T^{2} \)
13 \( 1 + (-0.134 - 0.0648i)T + (0.623 + 0.781i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.317 + 0.807i)T + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.974 + 1.68i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.988 - 0.149i)T^{2} \)
53 \( 1 + (-1.44 + 0.218i)T + (0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (-0.955 - 0.294i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.848 - 1.06i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (-0.563 - 0.975i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (1.57 + 1.07i)T + (0.365 + 0.930i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.641941001501302636478171375639, −8.151132741567267968363277566491, −7.16405824892932222808765290665, −6.18213301465150355194830227206, −5.58995107746465577183091496733, −4.71989832276323585220278609034, −3.86522262079375059033449756887, −2.76887513380615203704074058382, −2.28313805806652858038545625900, −0.58968798134225306611113522994, 0.70386911688408942165290017403, 2.89386413335424161572129003133, 3.41708209637661974034230151033, 4.80307353294412538620372935860, 5.40971153276064929437946273220, 5.80495982333538941891176969637, 6.72486949005264623225861689785, 7.43496926638173268958782244029, 8.149463566257131674153020005692, 8.825499334095506957422143118242

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