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Properties

Label 2-3332-476.331-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.271 + 0.962i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.607 − 0.465i)5-s + (0.707 + 0.707i)8-s + (0.965 − 0.258i)9-s + (−0.607 − 0.465i)10-s + (0.500 − 0.866i)16-s + (0.965 + 0.258i)17-s + (−0.499 − 0.866i)18-s + (−0.292 + 0.707i)20-s + (−0.107 + 0.400i)25-s + (0.707 + 0.292i)29-s + (−0.965 − 0.258i)32-s − i·34-s + (−0.707 + 0.707i)36-s + (0.465 + 0.607i)37-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.607 − 0.465i)5-s + (0.707 + 0.707i)8-s + (0.965 − 0.258i)9-s + (−0.607 − 0.465i)10-s + (0.500 − 0.866i)16-s + (0.965 + 0.258i)17-s + (−0.499 − 0.866i)18-s + (−0.292 + 0.707i)20-s + (−0.107 + 0.400i)25-s + (0.707 + 0.292i)29-s + (−0.965 − 0.258i)32-s − i·34-s + (−0.707 + 0.707i)36-s + (0.465 + 0.607i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.271 + 0.962i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.271 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.281646923\)
\(L(\frac12)\) \(\approx\) \(1.281646923\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 \)
17 \( 1 + (-0.965 - 0.258i)T \)
good3 \( 1 + (-0.965 + 0.258i)T^{2} \)
5 \( 1 + (-0.607 + 0.465i)T + (0.258 - 0.965i)T^{2} \)
11 \( 1 + (0.258 + 0.965i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.965 - 0.258i)T^{2} \)
29 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.965 + 0.258i)T^{2} \)
37 \( 1 + (-0.465 - 0.607i)T + (-0.258 + 0.965i)T^{2} \)
41 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.241 + 1.83i)T + (-0.965 - 0.258i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.0999 + 0.758i)T + (-0.965 + 0.258i)T^{2} \)
79 \( 1 + (-0.965 - 0.258i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
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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

−8.806101290480040599991616353423, −8.128031986114597876893872340909, −7.33531624530608161012821726323, −6.40273988801209504811531793809, −5.35896926635974758820591562181, −4.74668372420069027841849338373, −3.82671046431970182702685649021, −3.03258338465119621035031858123, −1.81659003137354326186323287432, −1.13907200535183769131176040800, 1.17118602317589879789733212688, 2.35909583238435316373018166071, 3.67041184292760459942877887781, 4.51309803123966080974126927089, 5.38111644843605994279388236134, 6.00185401305150571102613307721, 6.90501076092790324059239060004, 7.27164985563088754454314954644, 8.158290096828320320447379030736, 8.813232313571652924358442301855

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