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Properties

Label 2-2520-840.419-c0-0-6
Degree $2$
Conductor $2520$
Sign $0.985 - 0.169i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
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  + i·2-s − 4-s + 5-s + (−0.707 + 0.707i)7-s − i·8-s + i·10-s − 1.41i·11-s − 1.41i·13-s + (−0.707 − 0.707i)14-s + 16-s − 2i·19-s − 20-s + 1.41·22-s + 25-s + 1.41·26-s + ⋯
L(s)  = 1  + i·2-s − 4-s + 5-s + (−0.707 + 0.707i)7-s − i·8-s + i·10-s − 1.41i·11-s − 1.41i·13-s + (−0.707 − 0.707i)14-s + 16-s − 2i·19-s − 20-s + 1.41·22-s + 25-s + 1.41·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.985 - 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.109515298\)
\(L(\frac12)\) \(\approx\) \(1.109515298\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 2iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.012530589585873008748495374670, −8.471187037220825318148775721282, −7.55923204404627424014465428650, −6.55909757229210700322394973876, −6.04706989942097429454636523082, −5.48128965665231196585587785038, −4.74878534128852885253536371160, −3.26073181306445992822311906537, −2.73118706642937069888114534049, −0.77209736744654624123629054370, 1.54978834294404792650732720874, 2.08719710153022308030135648133, 3.34769739505761783909543713526, 4.19399561708656942074547006681, 4.87626276200714459789259096725, 5.97562496345165885433420405846, 6.68854182089306473605327832921, 7.58060465840117620284534641065, 8.576232287616946981504465802014, 9.587155834463209672418426436377

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