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Properties

Label 2-2520-5.4-c1-0-9
Degree $2$
Conductor $2520$
Sign $0.139 - 0.990i$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
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  + (0.311 − 2.21i)5-s + i·7-s + 5.05·11-s + 3.37i·13-s + 7.18i·17-s − 8.23·19-s + 6.23i·23-s + (−4.80 − 1.37i)25-s − 2·29-s − 4.62·31-s + (2.21 + 0.311i)35-s + 4.85i·37-s + 3.37·41-s + 1.24i·43-s − 49-s + ⋯
L(s)  = 1  + (0.139 − 0.990i)5-s + 0.377i·7-s + 1.52·11-s + 0.936i·13-s + 1.74i·17-s − 1.88·19-s + 1.30i·23-s + (−0.961 − 0.275i)25-s − 0.371·29-s − 0.830·31-s + (0.374 + 0.0525i)35-s + 0.798i·37-s + 0.527·41-s + 0.189i·43-s − 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.139 - 0.990i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ 0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.396737798\)
\(L(\frac12)\) \(\approx\) \(1.396737798\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-0.311 + 2.21i)T \)
7 \( 1 - iT \)
good11 \( 1 - 5.05T + 11T^{2} \)
13 \( 1 - 3.37iT - 13T^{2} \)
17 \( 1 - 7.18iT - 17T^{2} \)
19 \( 1 + 8.23T + 19T^{2} \)
23 \( 1 - 6.23iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 4.62T + 31T^{2} \)
37 \( 1 - 4.85iT - 37T^{2} \)
41 \( 1 - 3.37T + 41T^{2} \)
43 \( 1 - 1.24iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 4.62iT - 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 - 0.488T + 61T^{2} \)
67 \( 1 + 3.61iT - 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 16.2iT - 73T^{2} \)
79 \( 1 + 1.24T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 - 6.99T + 89T^{2} \)
97 \( 1 - 8.23iT - 97T^{2} \)
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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

−9.025152470979537827421871406606, −8.547139829394334220071075118496, −7.71160470258148473786824810273, −6.42398813866163903938370212069, −6.23072422175050269206776712279, −5.14134159132904275798642930051, −4.10816474842377458527301731847, −3.80411767320382444556712354270, −1.97071937623573987081472974222, −1.46958243388138610139944974824, 0.45390589846486834668214844741, 2.01301994673777986623677479661, 2.94241494771481543429428566107, 3.87004944542895542504601741244, 4.62883784707766816967180510744, 5.83968438960872854663851660703, 6.52206501372135568933127255217, 7.09221193310855699268141075409, 7.83770893285723335427696645378, 8.888222864051080494100210628142

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