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Properties

Label 2-1617-77.76-c1-0-17
Degree $2$
Conductor $1617$
Sign $-0.742 - 0.669i$
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  + 0.615i·2-s − i·3-s + 1.62·4-s + 3.28i·5-s + 0.615·6-s + 2.22i·8-s − 9-s − 2.02·10-s + (−0.0656 + 3.31i)11-s − 1.62i·12-s + 0.142·13-s + 3.28·15-s + 1.87·16-s − 7.26·17-s − 0.615i·18-s − 0.869·19-s + ⋯
L(s)  = 1  + 0.435i·2-s − 0.577i·3-s + 0.810·4-s + 1.46i·5-s + 0.251·6-s + 0.788i·8-s − 0.333·9-s − 0.639·10-s + (−0.0197 + 0.999i)11-s − 0.467i·12-s + 0.0395·13-s + 0.848·15-s + 0.467·16-s − 1.76·17-s − 0.145i·18-s − 0.199·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.742 - 0.669i$
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: \((2,\ 1617,\ (\ :1/2),\ -0.742 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.569604688\)
\(L(\frac12)\) \(\approx\) \(1.569604688\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0656 - 3.31i)T \)
good2 \( 1 - 0.615iT - 2T^{2} \)
5 \( 1 - 3.28iT - 5T^{2} \)
13 \( 1 - 0.142T + 13T^{2} \)
17 \( 1 + 7.26T + 17T^{2} \)
19 \( 1 + 0.869T + 19T^{2} \)
23 \( 1 + 3.56T + 23T^{2} \)
29 \( 1 - 3.70iT - 29T^{2} \)
31 \( 1 - 4.82iT - 31T^{2} \)
37 \( 1 - 9.39T + 37T^{2} \)
41 \( 1 + 3.19T + 41T^{2} \)
43 \( 1 + 9.66iT - 43T^{2} \)
47 \( 1 + 8.76iT - 47T^{2} \)
53 \( 1 + 6.62T + 53T^{2} \)
59 \( 1 - 1.54iT - 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 - 2.66T + 71T^{2} \)
73 \( 1 - 5.48T + 73T^{2} \)
79 \( 1 - 7.05iT - 79T^{2} \)
83 \( 1 - 6.31T + 83T^{2} \)
89 \( 1 - 7.87iT - 89T^{2} \)
97 \( 1 - 8.07iT - 97T^{2} \)
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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

−9.811553062138127586684992959386, −8.684356143011238016913252626758, −7.79526710689230100358632201310, −7.03879739701040112753651105786, −6.71500536742912170735681024287, −6.09007472654165641244643396858, −4.89417607736762393988751461599, −3.58193908935762236079862391306, −2.45396594304447504459314499807, −1.98279274601589004691109909012, 0.54165649687048734556918312696, 1.86209607102753946022519353587, 2.94804382213526458578929036902, 4.15111420768180260817261431133, 4.67243508115196179307364389284, 5.94676465723535935371361110276, 6.33461615263627596566987324660, 7.80548096113089553089518765730, 8.352655927923652850197433206390, 9.282830741484887341335305861698

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