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Properties

Label 2-1617-231.230-c0-0-7
Degree $2$
Conductor $1617$
Sign $0.727 + 0.686i$
Analytic cond. $0.806988$
Root an. cond. $0.898325$
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 − 4-s + 0.765·5-s + (0.707 − 0.707i)9-s − i·11-s + (−0.923 + 0.382i)12-s + (0.707 − 0.292i)15-s + 16-s − 0.765·20-s + 1.41i·23-s − 0.414·25-s + (0.382 − 0.923i)27-s − 1.84i·31-s + (−0.382 − 0.923i)33-s + (−0.707 + 0.707i)36-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s − 4-s + 0.765·5-s + (0.707 − 0.707i)9-s − i·11-s + (−0.923 + 0.382i)12-s + (0.707 − 0.292i)15-s + 16-s − 0.765·20-s + 1.41i·23-s − 0.414·25-s + (0.382 − 0.923i)27-s − 1.84i·31-s + (−0.382 − 0.923i)33-s + (−0.707 + 0.707i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(0.806988\)
Root analytic conductor: \(0.898325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (1616, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :0),\ 0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.404389896\)
\(L(\frac12)\) \(\approx\) \(1.404389896\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 \)
11 \( 1 + iT \)
good2 \( 1 + T^{2} \)
5 \( 1 - 0.765T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.84iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.84T + T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + 1.84T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.84T + T^{2} \)
97 \( 1 - 0.765iT - T^{2} \)
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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

−9.445352353042768994751720411396, −8.796294362523881114822340708786, −8.008719791682957672788850689254, −7.36687864028680204329183572780, −6.04070420929193575410878372675, −5.59913468880768676168273275053, −4.27006093545143893269826674692, −3.53625947160286554102738613630, −2.46935852947599026671019140124, −1.18673235420461493787558955875, 1.66514986920064005578736953532, 2.73529341319358607885538109656, 3.85873707667709053865299493903, 4.65923626215204843377008117733, 5.30305529024235126798237366300, 6.51515515620466213928512008210, 7.46908999373056913826182906201, 8.354697213984427160042435763582, 8.961723511350171365917462356723, 9.610805597228784342472343201063

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