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Properties

Label 2-1617-77.76-c1-0-52
Degree $2$
Conductor $1617$
Sign $0.977 + 0.209i$
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  + 1.03i·2-s − i·3-s + 0.934·4-s − 3.55i·5-s + 1.03·6-s + 3.02i·8-s − 9-s + 3.66·10-s + (0.693 + 3.24i)11-s − 0.934i·12-s + 4.92·13-s − 3.55·15-s − 1.25·16-s + 5.73·17-s − 1.03i·18-s − 2.80·19-s + ⋯
L(s)  = 1  + 0.729i·2-s − 0.577i·3-s + 0.467·4-s − 1.58i·5-s + 0.421·6-s + 1.07i·8-s − 0.333·9-s + 1.16·10-s + (0.208 + 0.977i)11-s − 0.269i·12-s + 1.36·13-s − 0.917·15-s − 0.314·16-s + 1.39·17-s − 0.243i·18-s − 0.644·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.977 + 0.209i$
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: \((2,\ 1617,\ (\ :1/2),\ 0.977 + 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.283902657\)
\(L(\frac12)\) \(\approx\) \(2.283902657\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.693 - 3.24i)T \)
good2 \( 1 - 1.03iT - 2T^{2} \)
5 \( 1 + 3.55iT - 5T^{2} \)
13 \( 1 - 4.92T + 13T^{2} \)
17 \( 1 - 5.73T + 17T^{2} \)
19 \( 1 + 2.80T + 19T^{2} \)
23 \( 1 - 7.37T + 23T^{2} \)
29 \( 1 + 9.68iT - 29T^{2} \)
31 \( 1 - 7.44iT - 31T^{2} \)
37 \( 1 - 0.0589T + 37T^{2} \)
41 \( 1 + 8.35T + 41T^{2} \)
43 \( 1 - 6.43iT - 43T^{2} \)
47 \( 1 + 8.08iT - 47T^{2} \)
53 \( 1 + 3.55T + 53T^{2} \)
59 \( 1 + 9.88iT - 59T^{2} \)
61 \( 1 + 6.54T + 61T^{2} \)
67 \( 1 - 5.87T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 4.35T + 73T^{2} \)
79 \( 1 + 6.11iT - 79T^{2} \)
83 \( 1 + 0.336T + 83T^{2} \)
89 \( 1 + 3.49iT - 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−9.043408017333666885719970906657, −8.328690903423165796220341432955, −7.915941225192592282643324961671, −6.90801607714201015340951038100, −6.22985773365371227897934031218, −5.34401156252413802101117764979, −4.73239638096606911359864008198, −3.41683954797181140362387009704, −1.90005044667451460115266064316, −1.08934753772371220167447092091, 1.24066634432566073502663992841, 2.69825754939935435816816887815, 3.37658984579378920986097756805, 3.76429494332732138461072539722, 5.44630198657299585029903492364, 6.30379705125606287224581124440, 6.83604040191809014263311194625, 7.80830644965457135486237992927, 8.815424161268583044738437932374, 9.683983646316825395469432447462

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