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Properties

Label 2-3332-68.67-c0-0-7
Degree $2$
Conductor $3332$
Sign $1$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 8-s − 11-s − 12-s − 13-s + 16-s + 17-s − 22-s + 2·23-s − 24-s + 25-s − 26-s + 27-s + 2·31-s + 32-s + 33-s + 34-s + 39-s − 44-s + 2·46-s − 48-s + 50-s − 51-s − 52-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 8-s − 11-s − 12-s − 13-s + 16-s + 17-s − 22-s + 2·23-s − 24-s + 25-s − 26-s + 27-s + 2·31-s + 32-s + 33-s + 34-s + 39-s − 44-s + 2·46-s − 48-s + 50-s − 51-s − 52-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3332} (883, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.688036655\)
\(L(\frac12)\) \(\approx\) \(1.688036655\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.672535400823614558709734307154, −7.77066689508039960791721988359, −7.09377876127807547519649891787, −6.40811280448717696616307850827, −5.58590977911947738300952748960, −4.97944284632773598745577165062, −4.62150966073764101748083850606, −3.07512903935494220247701262420, −2.71007625616584400486278396897, −1.07782926336385960956704110602, 1.07782926336385960956704110602, 2.71007625616584400486278396897, 3.07512903935494220247701262420, 4.62150966073764101748083850606, 4.97944284632773598745577165062, 5.58590977911947738300952748960, 6.40811280448717696616307850827, 7.09377876127807547519649891787, 7.77066689508039960791721988359, 8.672535400823614558709734307154

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