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Properties

Label 2-1617-77.76-c1-0-39
Degree $2$
Conductor $1617$
Sign $0.698 - 0.715i$
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  + 2.58i·2-s − i·3-s − 4.68·4-s + 1.50i·5-s + 2.58·6-s − 6.95i·8-s − 9-s − 3.88·10-s + (−0.276 − 3.30i)11-s + 4.68i·12-s − 3.99·13-s + 1.50·15-s + 8.61·16-s − 2.76·17-s − 2.58i·18-s + 4.91·19-s + ⋯
L(s)  = 1  + 1.82i·2-s − 0.577i·3-s − 2.34·4-s + 0.671i·5-s + 1.05·6-s − 2.45i·8-s − 0.333·9-s − 1.22·10-s + (−0.0832 − 0.996i)11-s + 1.35i·12-s − 1.10·13-s + 0.387·15-s + 2.15·16-s − 0.670·17-s − 0.609i·18-s + 1.12·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.698 - 0.715i$
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: \((2,\ 1617,\ (\ :1/2),\ 0.698 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.035239646\)
\(L(\frac12)\) \(\approx\) \(1.035239646\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.276 + 3.30i)T \)
good2 \( 1 - 2.58iT - 2T^{2} \)
5 \( 1 - 1.50iT - 5T^{2} \)
13 \( 1 + 3.99T + 13T^{2} \)
17 \( 1 + 2.76T + 17T^{2} \)
19 \( 1 - 4.91T + 19T^{2} \)
23 \( 1 + 0.0719T + 23T^{2} \)
29 \( 1 - 5.06iT - 29T^{2} \)
31 \( 1 + 10.2iT - 31T^{2} \)
37 \( 1 - 7.81T + 37T^{2} \)
41 \( 1 + 2.98T + 41T^{2} \)
43 \( 1 + 5.41iT - 43T^{2} \)
47 \( 1 + 5.18iT - 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 + 6.37iT - 59T^{2} \)
61 \( 1 + 1.37T + 61T^{2} \)
67 \( 1 + 4.55T + 67T^{2} \)
71 \( 1 - 9.59T + 71T^{2} \)
73 \( 1 - 8.96T + 73T^{2} \)
79 \( 1 + 11.6iT - 79T^{2} \)
83 \( 1 + 9.34T + 83T^{2} \)
89 \( 1 - 13.6iT - 89T^{2} \)
97 \( 1 + 2.63iT - 97T^{2} \)
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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

−9.167614673915457660172070065009, −8.392510533468072089909125197210, −7.63167139715856935845115731748, −7.10333764905486962668294453801, −6.45376124489614027370754751068, −5.65103063688131429674996705660, −4.98006701037769834152623869784, −3.77506885468392135203680527189, −2.58378947273334080408368288781, −0.49164280104290715068168691014, 1.04752009062973103701958078741, 2.28441662139748552623231070336, 3.06540054625018601726519997563, 4.27549453526815673249873283082, 4.71846243261408189043080123863, 5.41481461727753347054265045473, 7.02560963076157298047098762783, 8.120448454661266084697950497860, 8.989901873149939197493639208438, 9.591581752114386358075178170120

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