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Properties

Label 2-2520-7.2-c1-0-4
Degree $2$
Conductor $2520$
Sign $-0.701 - 0.712i$
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  + (−0.5 + 0.866i)5-s + (0.5 + 2.59i)7-s + (−1 − 1.73i)11-s + 13-s + (−1.5 + 2.59i)19-s + (−0.499 − 0.866i)25-s + (4.5 + 7.79i)31-s + (−2.5 − 0.866i)35-s + (−1.5 + 2.59i)37-s − 2·41-s + 3·43-s + (−3 + 5.19i)47-s + (−6.5 + 2.59i)49-s + 1.99·55-s + (−2 − 3.46i)59-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.301 − 0.522i)11-s + 0.277·13-s + (−0.344 + 0.596i)19-s + (−0.0999 − 0.173i)25-s + (0.808 + 1.39i)31-s + (−0.422 − 0.146i)35-s + (−0.246 + 0.427i)37-s − 0.312·41-s + 0.457·43-s + (−0.437 + 0.757i)47-s + (−0.928 + 0.371i)49-s + 0.269·55-s + (−0.260 − 0.450i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.066516671\)
\(L(\frac12)\) \(\approx\) \(1.066516671\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-4.5 - 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 3T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.046502553664370268660308140522, −8.401340311052111018721201385101, −7.86093704983684182060545729224, −6.79008653767653431209116169550, −6.10998724273518174755242839035, −5.37012095030339318983678769393, −4.47964789811145656219081102061, −3.35496191506312556476808847210, −2.66263514703228905682009739745, −1.46578173202486610092740303495, 0.35598933204592419380338541279, 1.59851258659846327415508034260, 2.80555058472077783504835140222, 4.00574083414178767156899211747, 4.48979605561166295090958934247, 5.40591509166606710614496311835, 6.40334328231317704171973469948, 7.20967667627442631268911740403, 7.81668569693202870741659074953, 8.555803318689080057763811484465

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