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Properties

Label 2-3332-3332.2923-c0-0-0
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 + (−1.82 − 0.563i)3-s + (−0.988 + 0.149i)4-s + (0.425 − 1.86i)6-s + (0.222 + 0.974i)7-s + (−0.222 − 0.974i)8-s + (2.19 + 1.49i)9-s + (−0.123 + 0.0841i)11-s + (1.88 + 0.284i)12-s + (−0.134 − 0.0648i)13-s + (−0.955 + 0.294i)14-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−1.32 + 2.29i)18-s + (0.142 − 1.90i)21-s + (−0.0931 − 0.116i)22-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)2-s + (−1.82 − 0.563i)3-s + (−0.988 + 0.149i)4-s + (0.425 − 1.86i)6-s + (0.222 + 0.974i)7-s + (−0.222 − 0.974i)8-s + (2.19 + 1.49i)9-s + (−0.123 + 0.0841i)11-s + (1.88 + 0.284i)12-s + (−0.134 − 0.0648i)13-s + (−0.955 + 0.294i)14-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−1.32 + 2.29i)18-s + (0.142 − 1.90i)21-s + (−0.0931 − 0.116i)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

\(L(\frac{1}{2})\) \(\approx\) \(0.5899183442\)
\(L(\frac12)\) \(\approx\) \(0.5899183442\)
\(L(1)\) 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 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.365 - 0.930i)T \)
good3 \( 1 + (1.82 + 0.563i)T + (0.826 + 0.563i)T^{2} \)
5 \( 1 + (-0.0747 + 0.997i)T^{2} \)
11 \( 1 + (0.123 - 0.0841i)T + (0.365 - 0.930i)T^{2} \)
13 \( 1 + (0.134 + 0.0648i)T + (0.623 + 0.781i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.658 + 1.67i)T + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.914 - 1.14i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (-0.826 - 1.43i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.676882442267301657814677411331, −8.147553521022779890387718856257, −7.25235969354013668226296304384, −6.51365563854547782335388930807, −6.13979552284261500320020296661, −5.32605200792473416004831162365, −4.91284949884844092799315555644, −4.02460923263171116028899104273, −2.33105082181338681546076180988, −0.919696082950570671571896365228, 0.64355780734717436148257273164, 1.57258342611680514737277280315, 3.32595382080973913998731687153, 3.96338913511966700290518231495, 4.89759936778790877662427857227, 5.23198014326493783744647303953, 6.03979516192960644747684883174, 7.13973759811321199765651432554, 7.62367832059738817214075124289, 9.107730744206378482766052446875

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