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Properties

Label 2-1632-408.371-c0-0-1
Degree $2$
Conductor $1632$
Sign $0.601 + 0.798i$
Analytic cond. $0.814474$
Root an. cond. $0.902482$
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  + (0.923 − 0.382i)3-s + (0.707 − 0.707i)9-s + (−0.382 − 1.92i)11-s + (0.923 − 0.382i)17-s + (−1.30 + 0.541i)19-s + (−0.382 + 0.923i)25-s + (0.382 − 0.923i)27-s + (−1.08 − 1.63i)33-s + (−0.216 + 0.324i)41-s + (1.70 + 0.707i)43-s + (0.382 + 0.923i)49-s + (0.707 − 0.707i)51-s + (−0.999 + i)57-s + (0.707 + 0.292i)59-s + 0.765i·67-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (0.707 − 0.707i)9-s + (−0.382 − 1.92i)11-s + (0.923 − 0.382i)17-s + (−1.30 + 0.541i)19-s + (−0.382 + 0.923i)25-s + (0.382 − 0.923i)27-s + (−1.08 − 1.63i)33-s + (−0.216 + 0.324i)41-s + (1.70 + 0.707i)43-s + (0.382 + 0.923i)49-s + (0.707 − 0.707i)51-s + (−0.999 + i)57-s + (0.707 + 0.292i)59-s + 0.765i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1632\)    =    \(2^{5} \cdot 3 \cdot 17\)
Sign: $0.601 + 0.798i$
Analytic conductor: \(0.814474\)
Root analytic conductor: \(0.902482\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1632} (1391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1632,\ (\ :0),\ 0.601 + 0.798i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.497275385\)
\(L(\frac12)\) \(\approx\) \(1.497275385\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
good5 \( 1 + (0.382 - 0.923i)T^{2} \)
7 \( 1 + (-0.382 - 0.923i)T^{2} \)
11 \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.923 + 0.382i)T^{2} \)
29 \( 1 + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.216 - 0.324i)T + (-0.382 - 0.923i)T^{2} \)
43 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.382 + 0.923i)T^{2} \)
67 \( 1 - 0.765iT - T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
79 \( 1 + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
97 \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \)
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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

−9.271121745318293096092191728614, −8.562709705528062218712914388410, −8.011972723062753600409544209948, −7.27235500409309485770448708177, −6.17208829761249096978808716136, −5.59883290885415566726096838648, −4.17940226406245292517747758144, −3.34327043915631861680166458816, −2.57041775391286396584463782847, −1.16086926993920994934769211498, 1.90844921756779610832750500764, 2.58710326822879219428261949851, 3.93803556562174473399307573636, 4.48514124356304196403219541084, 5.44165955189665913618670730239, 6.72458237558963090738164267542, 7.42325281427787093779957143849, 8.144103261098219344444522369384, 8.903785706994202587859978115015, 9.788248893289337474316043754370

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