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Properties

Label 2-3332-476.447-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.655 - 0.755i$
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  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (−1.63 − 0.324i)5-s + (−0.923 − 0.382i)8-s + (0.382 − 0.923i)9-s + (−0.324 − 1.63i)10-s + (−1 + i)13-s − i·16-s + (0.923 − 0.382i)17-s + 18-s + (1.38 − 0.923i)20-s + (1.63 + 0.675i)25-s + (−1.30 − 0.541i)26-s + (1.63 + 0.324i)29-s + (0.923 − 0.382i)32-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (−1.63 − 0.324i)5-s + (−0.923 − 0.382i)8-s + (0.382 − 0.923i)9-s + (−0.324 − 1.63i)10-s + (−1 + i)13-s − i·16-s + (0.923 − 0.382i)17-s + 18-s + (1.38 − 0.923i)20-s + (1.63 + 0.675i)25-s + (−1.30 − 0.541i)26-s + (1.63 + 0.324i)29-s + (0.923 − 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.655 - 0.755i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.655 - 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9520892500\)
\(L(\frac12)\) \(\approx\) \(0.9520892500\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 \)
17 \( 1 + (-0.923 + 0.382i)T \)
good3 \( 1 + (-0.382 + 0.923i)T^{2} \)
5 \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.382 + 0.923i)T^{2} \)
29 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (-1.92 + 0.382i)T + (0.923 - 0.382i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.0761 + 0.382i)T + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (-1.08 - 0.216i)T + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \)
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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

−8.823109206667194693202719294954, −7.922318879497863688501989508852, −7.33009259371484456636303457079, −6.93764564457216459236238502471, −5.97121923036487830064745367111, −4.91838261180442431730302125229, −4.32588279678546734717344980967, −3.75320897067519182747173004985, −2.84265920237594508674792660421, −0.75338268380273288994846534574, 0.873822057349656624008884387724, 2.41371966081223272242639332675, 3.14755882911110498273843616743, 3.93927756871623078463662209014, 4.73474772549740485192416813520, 5.26832140116316583273203288119, 6.44437371002927748019496787789, 7.49126400908310215419479802859, 7.963859740913043842546921787914, 8.539277637454586823114180111845

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