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Properties

Label 2-1323-147.11-c0-0-0
Degree $2$
Conductor $1323$
Sign $-0.180 + 0.983i$
Analytic cond. $0.660263$
Root an. cond. $0.812565$
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  + (0.0747 − 0.997i)4-s + (−0.222 − 0.974i)7-s + (−0.162 − 0.712i)13-s + (−0.988 − 0.149i)16-s + (0.222 + 0.385i)19-s + (−0.733 − 0.680i)25-s + (−0.988 + 0.149i)28-s + (−0.0747 + 0.129i)31-s + (0.0111 + 0.149i)37-s + (1.19 − 1.49i)43-s + (−0.900 + 0.433i)49-s + (−0.722 + 0.108i)52-s + (0.123 + 1.64i)61-s + (−0.222 + 0.974i)64-s + (0.988 − 1.71i)67-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)4-s + (−0.222 − 0.974i)7-s + (−0.162 − 0.712i)13-s + (−0.988 − 0.149i)16-s + (0.222 + 0.385i)19-s + (−0.733 − 0.680i)25-s + (−0.988 + 0.149i)28-s + (−0.0747 + 0.129i)31-s + (0.0111 + 0.149i)37-s + (1.19 − 1.49i)43-s + (−0.900 + 0.433i)49-s + (−0.722 + 0.108i)52-s + (0.123 + 1.64i)61-s + (−0.222 + 0.974i)64-s + (0.988 − 1.71i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.180 + 0.983i$
Analytic conductor: \(0.660263\)
Root analytic conductor: \(0.812565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ -0.180 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9650966089\)
\(L(\frac12)\) \(\approx\) \(0.9650966089\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.222 + 0.974i)T \)
good2 \( 1 + (-0.0747 + 0.997i)T^{2} \)
5 \( 1 + (0.733 + 0.680i)T^{2} \)
11 \( 1 + (-0.826 - 0.563i)T^{2} \)
13 \( 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.365 + 0.930i)T^{2} \)
19 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.365 - 0.930i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.0111 - 0.149i)T + (-0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (-1.19 + 1.49i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.988 + 0.149i)T^{2} \)
59 \( 1 + (0.733 - 0.680i)T^{2} \)
61 \( 1 + (-0.123 - 1.64i)T + (-0.988 + 0.149i)T^{2} \)
67 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-1.32 - 1.22i)T + (0.0747 + 0.997i)T^{2} \)
79 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.826 + 0.563i)T^{2} \)
97 \( 1 - 1.91T + T^{2} \)
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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

−9.843374813380338317128946635636, −8.929452935506238743584559261037, −7.86027748417921097735780622186, −7.13186061104414742010830892216, −6.23609286498714035631676994034, −5.49213496371417081139198326979, −4.52927006730274254081411325441, −3.56061507904456465480541524700, −2.19968774885279357180678175639, −0.821380339996888952947828142632, 2.05198414937931244931017363533, 2.96435512325056360074412012493, 3.94567773176881608575635814031, 4.94336860627420572620071287863, 5.98983480984804038034756631224, 6.84901626447994872351066157028, 7.67390138975381429611845220222, 8.449978820643668442303629419372, 9.232897913029808051548006225443, 9.734657709174157034235544277300

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