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Properties

Label 2-1617-77.76-c1-0-55
Degree $2$
Conductor $1617$
Sign $0.410 + 0.911i$
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  − 0.416i·2-s − i·3-s + 1.82·4-s − 0.478i·5-s − 0.416·6-s − 1.59i·8-s − 9-s − 0.199·10-s + (3.17 + 0.949i)11-s − 1.82i·12-s + 5.34·13-s − 0.478·15-s + 2.98·16-s − 0.246·17-s + 0.416i·18-s − 4.04·19-s + ⋯
L(s)  = 1  − 0.294i·2-s − 0.577i·3-s + 0.913·4-s − 0.213i·5-s − 0.170·6-s − 0.563i·8-s − 0.333·9-s − 0.0629·10-s + (0.958 + 0.286i)11-s − 0.527i·12-s + 1.48·13-s − 0.123·15-s + 0.747·16-s − 0.0597·17-s + 0.0981i·18-s − 0.927·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.410 + 0.911i$
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: \((2,\ 1617,\ (\ :1/2),\ 0.410 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.446331425\)
\(L(\frac12)\) \(\approx\) \(2.446331425\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.17 - 0.949i)T \)
good2 \( 1 + 0.416iT - 2T^{2} \)
5 \( 1 + 0.478iT - 5T^{2} \)
13 \( 1 - 5.34T + 13T^{2} \)
17 \( 1 + 0.246T + 17T^{2} \)
19 \( 1 + 4.04T + 19T^{2} \)
23 \( 1 - 1.08T + 23T^{2} \)
29 \( 1 - 9.25iT - 29T^{2} \)
31 \( 1 + 1.16iT - 31T^{2} \)
37 \( 1 - 4.46T + 37T^{2} \)
41 \( 1 + 9.10T + 41T^{2} \)
43 \( 1 + 2.69iT - 43T^{2} \)
47 \( 1 + 7.86iT - 47T^{2} \)
53 \( 1 + 0.485T + 53T^{2} \)
59 \( 1 + 1.10iT - 59T^{2} \)
61 \( 1 - 5.33T + 61T^{2} \)
67 \( 1 - 3.05T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 7.88T + 73T^{2} \)
79 \( 1 + 14.2iT - 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 - 8.20iT - 97T^{2} \)
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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

−8.985680655429565323299418766501, −8.622717901380768165700911063535, −7.53925489431319386416759864351, −6.60903736843878494492114870825, −6.41573852432173527071875005766, −5.24494594338335645663029115903, −3.94719815335275649546164525478, −3.14304482468191638706362364952, −1.88058755557229754608908810591, −1.13398536991444978459442206762, 1.32362930506305338896292220709, 2.64335361156383923474005906555, 3.60474641709072735573461138914, 4.44751212597681806793709675302, 5.74406913597296281862433815077, 6.32135313726655614653078892660, 6.89735973645774019013345851895, 8.122641038811351280798532078690, 8.593958183497953015054979083766, 9.530680336126912204877372572143

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