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Properties

Label 2-1520-1520.949-c0-0-2
Degree $2$
Conductor $1520$
Sign $-0.923 + 0.382i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + i·5-s − 1.00·6-s − 7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)10-s + (−0.707 − 0.707i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + i·17-s + (−0.707 − 0.707i)19-s − 1.00·20-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + i·5-s − 1.00·6-s − 7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)10-s + (−0.707 − 0.707i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + i·17-s + (−0.707 − 0.707i)19-s − 1.00·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9478524070\)
\(L(\frac12)\) \(\approx\) \(0.9478524070\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 - iT \)
19 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 - iT - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
59 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
61 \( 1 + (1 - i)T - iT^{2} \)
67 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−10.36255303725153006377284118073, −9.317219198421174635145347032468, −8.396168028957627900222530570232, −7.45161726377329776898305974410, −6.53572342182688081720041047928, −6.13064408880757008731042103850, −5.29817591971601660002270386466, −4.28215515055147612677755283227, −3.45934106448385248818883948117, −2.64924254415084733130458090432, 0.65590237446195701505751543417, 1.77219099610571688341477566349, 3.15358603093360951506923686433, 4.15123469513337605492671572794, 5.01235437572756240553957610413, 6.03241347662929252323357313824, 6.37316298488958726861927586889, 7.31712838557274640983214103289, 8.698809554458151858996038479109, 9.338498413175467148372589283468

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