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Properties

Label 2-2720-85.32-c0-0-1
Degree $2$
Conductor $2720$
Sign $0.459 + 0.888i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)9-s + 17-s − 1.00i·25-s + (−0.292 − 0.707i)29-s + (0.292 − 0.707i)37-s + (0.707 + 0.292i)41-s − 1.00·45-s + (0.707 − 0.707i)49-s + (−1.70 − 0.707i)61-s + (0.292 − 0.707i)73-s + 1.00i·81-s + (0.707 − 0.707i)85-s − 1.41·89-s + (1.70 + 0.707i)97-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)9-s + 17-s − 1.00i·25-s + (−0.292 − 0.707i)29-s + (0.292 − 0.707i)37-s + (0.707 + 0.292i)41-s − 1.00·45-s + (0.707 − 0.707i)49-s + (−1.70 − 0.707i)61-s + (0.292 − 0.707i)73-s + 1.00i·81-s + (0.707 − 0.707i)85-s − 1.41·89-s + (1.70 + 0.707i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $0.459 + 0.888i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (2497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :0),\ 0.459 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.301636962\)
\(L(\frac12)\) \(\approx\) \(1.301636962\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 - T \)
good3 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
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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

−9.010275844677760397645446412559, −8.208660635630575427419174876723, −7.47792877977125140025574570143, −6.34046820862361939966362875761, −5.83962440630297102516430190224, −5.12953626658676884019744865256, −4.13225829399400577584894745529, −3.17096857010273153402944905809, −2.13128483244165077460597921015, −0.882787275241658337645694108300, 1.53861038056048456882100192906, 2.63913918993594566978374111139, 3.26072896951726343698648942184, 4.49344094820702679988357547944, 5.54842528167378875570602128409, 5.87439772164043135637811713996, 6.91835119052292757729467797294, 7.59028517298205246142587065404, 8.360885656673130256729429951906, 9.220826298384702226769421247434

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