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Properties

Label 2-1617-231.230-c0-0-0
Degree $2$
Conductor $1617$
Sign $0.522 - 0.852i$
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  + (0.382 − 0.923i)3-s − 4-s − 1.84·5-s + (−0.707 − 0.707i)9-s + i·11-s + (−0.382 + 0.923i)12-s + (−0.707 + 1.70i)15-s + 16-s + 1.84·20-s + 1.41i·23-s + 2.41·25-s + (−0.923 + 0.382i)27-s + 0.765i·31-s + (0.923 + 0.382i)33-s + (0.707 + 0.707i)36-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s − 4-s − 1.84·5-s + (−0.707 − 0.707i)9-s + i·11-s + (−0.382 + 0.923i)12-s + (−0.707 + 1.70i)15-s + 16-s + 1.84·20-s + 1.41i·23-s + 2.41·25-s + (−0.923 + 0.382i)27-s + 0.765i·31-s + (0.923 + 0.382i)33-s + (0.707 + 0.707i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.522 - 0.852i$
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: \((2,\ 1617,\ (\ :0),\ 0.522 - 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4180165346\)
\(L(\frac12)\) \(\approx\) \(0.4180165346\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.380591896319209505741223845664, −8.804525602491751492940104175216, −7.965341024197123946610373917772, −7.54251358709226031899407238130, −6.88934308678587642194527874457, −5.55674934777252227372014570269, −4.52371132224014815272402987601, −3.82723494919371461832199544457, −2.97793189577574573676263252433, −1.25958347742526314017455692611, 0.36882363803313883345811469357, 2.95184754439573166139906238177, 3.67879044278980547135239229625, 4.32577950616401072572894047412, 4.93803467299054265447728666740, 6.05930510598648180659862073714, 7.43100765121624480765578391683, 8.200739044720917557862027956708, 8.566619964052382016539972206601, 9.232157348995870370242072783801

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