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Properties

Label 2-3332-3332.1971-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.801 + 0.598i$
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  + (0.733 − 0.680i)2-s + (0.294 − 0.0444i)3-s + (0.0747 − 0.997i)4-s + (0.185 − 0.233i)6-s + (0.781 − 0.623i)7-s + (−0.623 − 0.781i)8-s + (−0.870 + 0.268i)9-s + (−1.29 − 0.400i)11-s + (−0.0222 − 0.297i)12-s + (−0.326 − 1.42i)13-s + (0.149 − 0.988i)14-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (−0.455 + 0.789i)18-s + (0.202 − 0.218i)21-s + (−1.22 + 0.590i)22-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.294 − 0.0444i)3-s + (0.0747 − 0.997i)4-s + (0.185 − 0.233i)6-s + (0.781 − 0.623i)7-s + (−0.623 − 0.781i)8-s + (−0.870 + 0.268i)9-s + (−1.29 − 0.400i)11-s + (−0.0222 − 0.297i)12-s + (−0.326 − 1.42i)13-s + (0.149 − 0.988i)14-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (−0.455 + 0.789i)18-s + (0.202 − 0.218i)21-s + (−1.22 + 0.590i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.801 + 0.598i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.801 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.756358729\)
\(L(\frac12)\) \(\approx\) \(1.756358729\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.733 + 0.680i)T \)
7 \( 1 + (-0.781 + 0.623i)T \)
17 \( 1 + (-0.826 + 0.563i)T \)
good3 \( 1 + (-0.294 + 0.0444i)T + (0.955 - 0.294i)T^{2} \)
5 \( 1 + (0.733 + 0.680i)T^{2} \)
11 \( 1 + (1.29 + 0.400i)T + (0.826 + 0.563i)T^{2} \)
13 \( 1 + (0.326 + 1.42i)T + (-0.900 + 0.433i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.61 - 1.09i)T + (0.365 + 0.930i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.781 - 1.35i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.988 - 0.149i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.0747 + 0.997i)T^{2} \)
53 \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.733 - 0.680i)T^{2} \)
61 \( 1 + (0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.67 + 0.807i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.0747 - 0.997i)T^{2} \)
79 \( 1 + (0.294 + 0.510i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-1.88 + 0.582i)T + (0.826 - 0.563i)T^{2} \)
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

−8.414215914752464600715475529059, −7.77555323832300806656112459490, −7.17944306504826164807210930712, −5.82901349478134368267909058792, −5.18961737113302489315932100795, −4.96092224739470216469176086226, −3.40098253290197017676696325720, −3.11227707907474474955394302873, −2.09543318601966574753378176228, −0.75281708539361017126704520878, 2.09650532055160615195493078363, 2.70043378715972056883347769835, 3.75502151397471017423932353466, 4.67767455482159794741944164178, 5.32607091770622523392743490995, 5.91829963951375480858887904375, 6.86715713257095363834036002697, 7.69103101291960758305778241672, 8.144283013952964666500262578281, 8.968893554813934164594052186964

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