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Properties

Label 2-3332-68.19-c0-0-1
Degree $2$
Conductor $3332$
Sign $-0.673 - 0.739i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 + 1.70i)5-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.707 + 1.70i)10-s − 1.00·16-s + (0.707 + 0.707i)17-s + 1.00·18-s + (−1.70 + 0.707i)20-s + (−1.70 + 1.70i)25-s + (−0.707 − 1.70i)29-s + (−0.707 − 0.707i)32-s + 1.00i·34-s + (0.707 + 0.707i)36-s + (1.70 − 0.707i)37-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 + 1.70i)5-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.707 + 1.70i)10-s − 1.00·16-s + (0.707 + 0.707i)17-s + 1.00·18-s + (−1.70 + 0.707i)20-s + (−1.70 + 1.70i)25-s + (−0.707 − 1.70i)29-s + (−0.707 − 0.707i)32-s + 1.00i·34-s + (0.707 + 0.707i)36-s + (1.70 − 0.707i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.673 - 0.739i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.673 - 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.141904986\)
\(L(\frac12)\) \(\approx\) \(2.141904986\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 \)
17 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
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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

−9.161511563119343540606500518318, −7.82562884592229213650325780499, −7.55424879790955622648732296622, −6.57032220503463674307292971235, −6.22901000901665781839665755070, −5.63198612189880790730157045095, −4.35716776850597121711549830772, −3.62707696532172022020501322927, −2.90474165750137057727457606987, −1.94827965416770219397599279817, 1.11342710516643518288797401479, 1.73998008984554966297433211330, 2.83461457848544409023271900029, 4.05975409333704124869233484045, 4.75249500590844874976955506936, 5.26899848961023729087418598942, 5.85304469049485541004284379128, 6.93437253246052987013893522412, 7.88717880989730488182322942049, 8.752627347151423413526107956341

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