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Properties

Label 2-1617-77.76-c1-0-9
Degree $2$
Conductor $1617$
Sign $0.375 + 0.926i$
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  + 1.89i·2-s + i·3-s − 1.58·4-s + 1.86i·5-s − 1.89·6-s + 0.793i·8-s − 9-s − 3.52·10-s + (−1.71 − 2.84i)11-s − 1.58i·12-s + 0.900·13-s − 1.86·15-s − 4.66·16-s − 4.81·17-s − 1.89i·18-s − 0.291·19-s + ⋯
L(s)  = 1  + 1.33i·2-s + 0.577i·3-s − 0.790·4-s + 0.834i·5-s − 0.772·6-s + 0.280i·8-s − 0.333·9-s − 1.11·10-s + (−0.516 − 0.856i)11-s − 0.456i·12-s + 0.249·13-s − 0.481·15-s − 1.16·16-s − 1.16·17-s − 0.446i·18-s − 0.0667·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.375 + 0.926i$
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: \((2,\ 1617,\ (\ :1/2),\ 0.375 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5542872684\)
\(L(\frac12)\) \(\approx\) \(0.5542872684\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 2.84i)T \)
good2 \( 1 - 1.89iT - 2T^{2} \)
5 \( 1 - 1.86iT - 5T^{2} \)
13 \( 1 - 0.900T + 13T^{2} \)
17 \( 1 + 4.81T + 17T^{2} \)
19 \( 1 + 0.291T + 19T^{2} \)
23 \( 1 + 3.81T + 23T^{2} \)
29 \( 1 + 3.00iT - 29T^{2} \)
31 \( 1 - 2.50iT - 31T^{2} \)
37 \( 1 + 7.22T + 37T^{2} \)
41 \( 1 + 3.97T + 41T^{2} \)
43 \( 1 + 8.50iT - 43T^{2} \)
47 \( 1 - 11.5iT - 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 - 8.44iT - 59T^{2} \)
61 \( 1 - 2.88T + 61T^{2} \)
67 \( 1 - 1.24T + 67T^{2} \)
71 \( 1 + 2.01T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 1.26iT - 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 - 7.48iT - 97T^{2} \)
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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

−9.963503997770911554502845341727, −8.884871433840301530570888673497, −8.422878342984110466659584870473, −7.56098121264508234237790184221, −6.71641256511651137130114461809, −6.14905050450935447613570724344, −5.35710553581698772187473160813, −4.45619922759882499105036323392, −3.35309653423043844908529883966, −2.31920398774584457928169514607, 0.19818693597120692558195207628, 1.57876745645785243744565225481, 2.22019830453399976704158344782, 3.37610279413032516222440373181, 4.43805561423164251499149648810, 5.11913948687161319859875940594, 6.41623077323389959315966278571, 7.13640802773474023587176518871, 8.220983261038394119595112119748, 8.874143157626212484868298453687

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