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Properties

Label 2-3332-476.179-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.806 - 0.591i$
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  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.758 − 0.0999i)5-s + (0.707 + 0.707i)8-s + (0.258 − 0.965i)9-s + (0.758 + 0.0999i)10-s + 2i·13-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)17-s + (0.499 − 0.866i)18-s + (0.707 + 0.292i)20-s + (−0.400 + 0.107i)25-s + (−0.517 + 1.93i)26-s + (0.707 − 1.70i)29-s + (0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.758 − 0.0999i)5-s + (0.707 + 0.707i)8-s + (0.258 − 0.965i)9-s + (0.758 + 0.0999i)10-s + 2i·13-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)17-s + (0.499 − 0.866i)18-s + (0.707 + 0.292i)20-s + (−0.400 + 0.107i)25-s + (−0.517 + 1.93i)26-s + (0.707 − 1.70i)29-s + (0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.806 - 0.591i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.806 - 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.692005085\)
\(L(\frac12)\) \(\approx\) \(2.692005085\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 \)
17 \( 1 + (0.965 - 0.258i)T \)
good3 \( 1 + (-0.258 + 0.965i)T^{2} \)
5 \( 1 + (-0.758 + 0.0999i)T + (0.965 - 0.258i)T^{2} \)
11 \( 1 + (0.965 + 0.258i)T^{2} \)
13 \( 1 - 2iT - T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.258 - 0.965i)T^{2} \)
29 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (-0.258 + 0.965i)T^{2} \)
37 \( 1 + (-0.241 - 1.83i)T + (-0.965 + 0.258i)T^{2} \)
41 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-1.12 + 1.46i)T + (-0.258 - 0.965i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.465 + 0.607i)T + (-0.258 + 0.965i)T^{2} \)
79 \( 1 + (-0.258 - 0.965i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
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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

−8.873811472280372881095851802659, −8.101054669267844377259889957447, −6.90639335400925480739958621389, −6.57266628247347888243948442885, −6.04665379802282139333375651515, −4.97374501679815143037461070888, −4.26364126922692552500892324421, −3.63309268528160370259010312089, −2.34005726015665350144263961367, −1.69873463642145819251897627896, 1.34992039060709542781108489212, 2.45105361586228812973313585210, 2.99095660707273757757103185042, 4.15209128887412439121202062442, 5.06001025624547660450227992198, 5.51231937884701399062257488877, 6.25637029570637513575254162380, 7.15690642581364787165535350796, 7.79376192424673425082823103348, 8.693144283792042496189032050584

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