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Properties

Label 2-3332-17.16-c1-0-16
Degree $2$
Conductor $3332$
Sign $-0.840 - 0.542i$
Analytic cond. $26.6061$
Root an. cond. $5.15811$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61i·3-s − 0.381i·5-s − 3.85·9-s + 3.46i·11-s + 5.60·13-s + 15-s + (−3.46 − 2.23i)17-s + 2.14·19-s + 2.14i·23-s + 4.85·25-s − 2.23i·27-s + 7.74i·29-s + 4.85i·31-s − 9.06·33-s − 9.06i·37-s + ⋯
L(s)  = 1  + 1.51i·3-s − 0.170i·5-s − 1.28·9-s + 1.04i·11-s + 1.55·13-s + 0.258·15-s + (−0.840 − 0.542i)17-s + 0.491·19-s + 0.446i·23-s + 0.970·25-s − 0.430i·27-s + 1.43i·29-s + 0.871i·31-s − 1.57·33-s − 1.49i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.840 - 0.542i$
Analytic conductor: \(26.6061\)
Root analytic conductor: \(5.15811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :1/2),\ -0.840 - 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.780961658\)
\(L(\frac12)\) \(\approx\) \(1.780961658\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 + (3.46 + 2.23i)T \)
good3 \( 1 - 2.61iT - 3T^{2} \)
5 \( 1 + 0.381iT - 5T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 - 5.60T + 13T^{2} \)
19 \( 1 - 2.14T + 19T^{2} \)
23 \( 1 - 2.14iT - 23T^{2} \)
29 \( 1 - 7.74iT - 29T^{2} \)
31 \( 1 - 4.85iT - 31T^{2} \)
37 \( 1 + 9.06iT - 37T^{2} \)
41 \( 1 + 2.61iT - 41T^{2} \)
43 \( 1 - 5.85T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 4.85T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 11.5iT - 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 - 9.06iT - 71T^{2} \)
73 \( 1 + 4.14iT - 73T^{2} \)
79 \( 1 - 7.74iT - 79T^{2} \)
83 \( 1 + 9.88T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 1.14iT - 97T^{2} \)
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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

−8.907726294747932747935762122715, −8.669605116930259278263584191342, −7.36761549484760061862415102720, −6.73440119696524537083701368101, −5.58294461599859872533555127501, −5.06493583889182356400676604395, −4.23748265203578384865544958956, −3.67043224649599321921372115004, −2.71849793775440241749226796441, −1.32086829240605591844262954489, 0.59658060743173226785314336543, 1.45511767795379466034053490702, 2.50736718298712108298920679475, 3.37536053348820213565971237135, 4.38568041309871618165133682709, 5.67615600924783379631930167456, 6.35877975712845526383285216263, 6.56897095048098749280076734846, 7.66158811000452807008000794917, 8.309111664128255137391447771435

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