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Properties

Label 2-4080-17.16-c1-0-40
Degree $2$
Conductor $4080$
Sign $0.242 + 0.970i$
Analytic cond. $32.5789$
Root an. cond. $5.70779$
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  − i·3-s − i·5-s − i·7-s − 9-s + 5i·11-s − 2·13-s − 15-s + (1 + 4i)17-s − 19-s − 21-s − 25-s + i·27-s − 7i·29-s + 5·33-s − 35-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447i·5-s − 0.377i·7-s − 0.333·9-s + 1.50i·11-s − 0.554·13-s − 0.258·15-s + (0.242 + 0.970i)17-s − 0.229·19-s − 0.218·21-s − 0.200·25-s + 0.192i·27-s − 1.29i·29-s + 0.870·33-s − 0.169·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4080\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(32.5789\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4080} (3841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4080,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.594885255\)
\(L(\frac12)\) \(\approx\) \(1.594885255\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + iT \)
17 \( 1 + (-1 - 4i)T \)
good7 \( 1 + iT - 7T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 7iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 5iT - 37T^{2} \)
41 \( 1 + 9iT - 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.123211971188026849455338291255, −7.47790825982868040208228288298, −7.02404288335124198733104987461, −6.06823521703467764431697250537, −5.36311432697206404917736402633, −4.38466077427138350371865489792, −3.88210362884565462898388751716, −2.41553158333585497620382978142, −1.84309156241406501852649650331, −0.56831208883210274896897443058, 0.911748960445109706226060729316, 2.53811271022661788320714090633, 3.06365004603294013069594368517, 3.92799505878477115689008770822, 4.94079703149127181163409864643, 5.57362264512281649208051253560, 6.28502812885181009240371287217, 7.12298792220899568091724321198, 7.919499707693284992677893705757, 8.726826360762679320933376006324

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