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Properties

Label 2-7524-209.208-c1-0-69
Degree $2$
Conductor $7524$
Sign $i$
Analytic cond. $60.0794$
Root an. cond. $7.75109$
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  − 3.31i·11-s − 2.79·13-s + 4.88i·17-s + 4.35·19-s − 5·25-s + 7·49-s − 8.72i·53-s − 2.72i·59-s − 14.7i·71-s + 17.1·79-s + 1.75i·83-s + 3.27i·89-s + 13.2i·101-s − 17.4·109-s − 20.7i·113-s + ⋯
L(s)  = 1  − 1.00i·11-s − 0.775·13-s + 1.18i·17-s + 1.00·19-s − 25-s + 49-s − 1.19i·53-s − 0.355i·59-s − 1.74i·71-s + 1.92·79-s + 0.192i·83-s + 0.346i·89-s + 1.32i·101-s − 1.67·109-s − 1.94i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7524\)    =    \(2^{2} \cdot 3^{2} \cdot 11 \cdot 19\)
Sign: $i$
Analytic conductor: \(60.0794\)
Root analytic conductor: \(7.75109\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7524} (2089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7524,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.320342450\)
\(L(\frac12)\) \(\approx\) \(1.320342450\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + 3.31iT \)
19 \( 1 - 4.35T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 + 2.79T + 13T^{2} \)
17 \( 1 - 4.88iT - 17T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 8.72iT - 53T^{2} \)
59 \( 1 + 2.72iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 14.7iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 - 1.75iT - 83T^{2} \)
89 \( 1 - 3.27iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−7.932357598514393710540418701146, −7.00528335766621916188209368420, −6.28212979488752551640863389536, −5.61217211956235493564289530179, −5.01878147752781845487843716308, −3.97867928390240849805444290673, −3.42695994872383856084380387440, −2.48852958374385687561246549992, −1.54080482326575401819105250747, −0.35544010580295572867287398026, 0.975413134280614393802005143722, 2.14641450627228403128946268609, 2.78408458280605220181841940106, 3.78729339707179229432266287478, 4.60415017282693730568558099230, 5.18871050149975344473231467998, 5.88642044267051179172700670178, 6.88966036490446343580110387190, 7.38390338720132679826217221268, 7.80844506687874279226971656608

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