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Properties

Label 2-2720-85.77-c0-0-1
Degree $2$
Conductor $2720$
Sign $0.825 - 0.564i$
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  + 5-s + (−0.707 + 0.707i)9-s + i·17-s + 25-s + (1.70 + 0.707i)29-s + (0.707 + 1.70i)37-s + (−0.707 − 1.70i)41-s + (−0.707 + 0.707i)45-s + (0.707 + 0.707i)49-s + (−0.292 − 0.707i)61-s + (−0.707 − 1.70i)73-s − 1.00i·81-s + i·85-s − 1.41·89-s + (−0.707 + 0.292i)97-s + ⋯
L(s)  = 1  + 5-s + (−0.707 + 0.707i)9-s + i·17-s + 25-s + (1.70 + 0.707i)29-s + (0.707 + 1.70i)37-s + (−0.707 − 1.70i)41-s + (−0.707 + 0.707i)45-s + (0.707 + 0.707i)49-s + (−0.292 − 0.707i)61-s + (−0.707 − 1.70i)73-s − 1.00i·81-s + i·85-s − 1.41·89-s + (−0.707 + 0.292i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $0.825 - 0.564i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :0),\ 0.825 - 0.564i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.397884951\)
\(L(\frac12)\) \(\approx\) \(1.397884951\)
\(L(1)\) 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 \( 1 - T \)
17 \( 1 - iT \)
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 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
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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

−8.890444149952133524939383502231, −8.525244734265594132577717677777, −7.64554737562890441595990134533, −6.60933676677434535929817560393, −6.05322691876714162247291874712, −5.24324856974396468083967219713, −4.55778385258899669629483587980, −3.25371566233854301031448321883, −2.43290854611006110592312550452, −1.44005639849379349465709122429, 0.989736599639706829070217833027, 2.41856593008167393071279856405, 3.02495002883602578799739959943, 4.25562730228362038907602769009, 5.17718618850907817026705127049, 5.92139873378589822629124520942, 6.52730028796287571832979352493, 7.32100076912681053769279825539, 8.395504159502385641473322670605, 8.951477574332714184166493637776

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