Properties

Label 2-1520-5.4-c1-0-48
Degree $2$
Conductor $1520$
Sign $-0.948 + 0.316i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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  − 2.41i·3-s + (2.12 − 0.707i)5-s − 2.41i·7-s − 2.82·9-s − 1.41·11-s + 1.82i·13-s + (−1.70 − 5.12i)15-s i·17-s − 19-s − 5.82·21-s − 5.24i·23-s + (3.99 − 3i)25-s − 0.414i·27-s − 3.82·29-s − 3.41·31-s + ⋯
L(s)  = 1  − 1.39i·3-s + (0.948 − 0.316i)5-s − 0.912i·7-s − 0.942·9-s − 0.426·11-s + 0.507i·13-s + (−0.440 − 1.32i)15-s − 0.242i·17-s − 0.229·19-s − 1.27·21-s − 1.09i·23-s + (0.799 − 0.600i)25-s − 0.0797i·27-s − 0.710·29-s − 0.613·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.948 + 0.316i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ -0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.670319909\)
\(L(\frac12)\) \(\approx\) \(1.670319909\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.12 + 0.707i)T \)
19 \( 1 + T \)
good3 \( 1 + 2.41iT - 3T^{2} \)
7 \( 1 + 2.41iT - 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 - 1.82iT - 13T^{2} \)
17 \( 1 + iT - 17T^{2} \)
23 \( 1 + 5.24iT - 23T^{2} \)
29 \( 1 + 3.82T + 29T^{2} \)
31 \( 1 + 3.41T + 31T^{2} \)
37 \( 1 + 5.17iT - 37T^{2} \)
41 \( 1 + 7.07T + 41T^{2} \)
43 \( 1 + 2.24iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 1.82iT - 53T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 - 3.41T + 61T^{2} \)
67 \( 1 + 6.07iT - 67T^{2} \)
71 \( 1 + 3.07T + 71T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 + 0.828T + 79T^{2} \)
83 \( 1 - 2.48iT - 83T^{2} \)
89 \( 1 - 3.75T + 89T^{2} \)
97 \( 1 + 7.65iT - 97T^{2} \)
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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

−8.977997089393732275069436107736, −8.226114675530389908845938839060, −7.30023868640796392684681351291, −6.81818591236742531507530799066, −6.02828787343619651952209657033, −5.11127440523543769365541882968, −4.00902587571035455570869971860, −2.52657008241130813305408273596, −1.74327967522851569861427311134, −0.63338995933067839325469605499, 1.91332638754287278901993752509, 2.98928491061079411314799678318, 3.80605261575253879895754112899, 5.17451060186451044927248202297, 5.38197193791193878841825432983, 6.31443354785152808386557464951, 7.45474023547109487275819863749, 8.627493739726744200707091368070, 9.115092753433907488094562286481, 10.00625112711815264058368887428

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