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Properties

Label 2-1617-77.76-c1-0-5
Degree $2$
Conductor $1617$
Sign $-0.920 - 0.391i$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
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  − 1.38i·2-s + i·3-s + 0.0776·4-s + 3.79i·5-s + 1.38·6-s − 2.88i·8-s − 9-s + 5.26·10-s + (−2.81 − 1.76i)11-s + 0.0776i·12-s − 6.27·13-s − 3.79·15-s − 3.83·16-s − 3.89·17-s + 1.38i·18-s + 5.19·19-s + ⋯
L(s)  = 1  − 0.980i·2-s + 0.577i·3-s + 0.0388·4-s + 1.69i·5-s + 0.566·6-s − 1.01i·8-s − 0.333·9-s + 1.66·10-s + (−0.847 − 0.530i)11-s + 0.0224i·12-s − 1.73·13-s − 0.981·15-s − 0.959·16-s − 0.945·17-s + 0.326i·18-s + 1.19·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.920 - 0.391i$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ -0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2257211731\)
\(L(\frac12)\) \(\approx\) \(0.2257211731\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 \)
11 \( 1 + (2.81 + 1.76i)T \)
good2 \( 1 + 1.38iT - 2T^{2} \)
5 \( 1 - 3.79iT - 5T^{2} \)
13 \( 1 + 6.27T + 13T^{2} \)
17 \( 1 + 3.89T + 17T^{2} \)
19 \( 1 - 5.19T + 19T^{2} \)
23 \( 1 - 1.05T + 23T^{2} \)
29 \( 1 + 1.48iT - 29T^{2} \)
31 \( 1 - 4.41iT - 31T^{2} \)
37 \( 1 + 0.0249T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 7.27iT - 43T^{2} \)
47 \( 1 + 6.12iT - 47T^{2} \)
53 \( 1 - 0.812T + 53T^{2} \)
59 \( 1 + 6.48iT - 59T^{2} \)
61 \( 1 + 8.28T + 61T^{2} \)
67 \( 1 - 4.35T + 67T^{2} \)
71 \( 1 + 9.74T + 71T^{2} \)
73 \( 1 - 5.28T + 73T^{2} \)
79 \( 1 - 4.42iT - 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 - 18.2iT - 89T^{2} \)
97 \( 1 + 19.4iT - 97T^{2} \)
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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

−9.971951301324964529907765830917, −9.493157802763500314314467795437, −8.129343536935809875540014351966, −7.13454793820528928053590669063, −6.78105797098600897286396835641, −5.57517361452779019812254386322, −4.56995218450128379621560715739, −3.27989808591473276932236723272, −2.93918511846356978737726493232, −2.11359073800950229220949896481, 0.07601701342469189460593665036, 1.72174384142879571859122683725, 2.66430700464528606147761456011, 4.54728357772176767142111224359, 5.11276499028227796433831720840, 5.63784163647512221472797160015, 6.84904267355569929756004620378, 7.50219746548622778835593685052, 7.998306848954985056338692534817, 8.859493652334221521566817414840

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