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Properties

Label 2-3168-88.43-c1-0-35
Degree $2$
Conductor $3168$
Sign $0.996 + 0.0829i$
Analytic cond. $25.2966$
Root an. cond. $5.02957$
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  + 2.44·7-s + (3 − 1.41i)11-s + 4.89·13-s + 7.07i·17-s − 4.24i·19-s − 6.92i·23-s + 5·25-s − 2.44·29-s + 10.3i·31-s − 3.46i·37-s − 7.07i·41-s − 4.24i·43-s + 10.3i·47-s − 1.00·49-s − 6·59-s + ⋯
L(s)  = 1  + 0.925·7-s + (0.904 − 0.426i)11-s + 1.35·13-s + 1.71i·17-s − 0.973i·19-s − 1.44i·23-s + 25-s − 0.454·29-s + 1.86i·31-s − 0.569i·37-s − 1.10i·41-s − 0.646i·43-s + 1.51i·47-s − 0.142·49-s − 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3168\)    =    \(2^{5} \cdot 3^{2} \cdot 11\)
Sign: $0.996 + 0.0829i$
Analytic conductor: \(25.2966\)
Root analytic conductor: \(5.02957\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3168} (2287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3168,\ (\ :1/2),\ 0.996 + 0.0829i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.462514284\)
\(L(\frac12)\) \(\approx\) \(2.462514284\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (-3 + 1.41i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 2.44T + 7T^{2} \)
13 \( 1 - 4.89T + 13T^{2} \)
17 \( 1 - 7.07iT - 17T^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 + 6.92iT - 23T^{2} \)
29 \( 1 + 2.44T + 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 + 3.46iT - 37T^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 9.79T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 2.44T + 79T^{2} \)
83 \( 1 + 5.65iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−8.724912914627862228991610483137, −8.185409790056709939892369437858, −7.05085132150502945904957714745, −6.41845795277355787100499400798, −5.71848071561613727451902823187, −4.70571902775812438424484987148, −4.01365294742535125480792458361, −3.15848068301747871904406934433, −1.83228422071514076277895722157, −1.02357803281580801932460598546, 1.06515969058692050244643648720, 1.86291904828081536791279261673, 3.17758220875247705377730357964, 4.00725900276259756110602620704, 4.81876499274502794905529308995, 5.63226615351068712772797279640, 6.41959657282865727743784929368, 7.28950343394420670626352785488, 7.922009955455695563278560499934, 8.663933081965159234833987828899

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