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Properties

Label 2-1480-37.10-c1-0-31
Degree $2$
Conductor $1480$
Sign $-0.773 - 0.634i$
Analytic cond. $11.8178$
Root an. cond. $3.43771$
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  + (−1.16 − 2.01i)3-s + (0.5 + 0.866i)5-s + (0.511 + 0.886i)7-s + (−1.20 + 2.07i)9-s − 3.07·11-s + (1.33 + 2.30i)13-s + (1.16 − 2.01i)15-s + (1.83 − 3.17i)17-s + (−0.553 − 0.958i)19-s + (1.18 − 2.05i)21-s − 5.96·23-s + (−0.499 + 0.866i)25-s − 1.39·27-s − 0.619·29-s − 6.18·31-s + ⋯
L(s)  = 1  + (−0.670 − 1.16i)3-s + (0.223 + 0.387i)5-s + (0.193 + 0.334i)7-s + (−0.400 + 0.693i)9-s − 0.928·11-s + (0.369 + 0.639i)13-s + (0.300 − 0.519i)15-s + (0.444 − 0.769i)17-s + (−0.126 − 0.219i)19-s + (0.259 − 0.449i)21-s − 1.24·23-s + (−0.0999 + 0.173i)25-s − 0.267·27-s − 0.114·29-s − 1.11·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.773 - 0.634i$
Analytic conductor: \(11.8178\)
Root analytic conductor: \(3.43771\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1480,\ (\ :1/2),\ -0.773 - 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04798784706\)
\(L(\frac12)\) \(\approx\) \(0.04798784706\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (5.94 - 1.28i)T \)
good3 \( 1 + (1.16 + 2.01i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.511 - 0.886i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.07T + 11T^{2} \)
13 \( 1 + (-1.33 - 2.30i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.83 + 3.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.553 + 0.958i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.96T + 23T^{2} \)
29 \( 1 + 0.619T + 29T^{2} \)
31 \( 1 + 6.18T + 31T^{2} \)
41 \( 1 + (4.79 + 8.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 2.03T + 43T^{2} \)
47 \( 1 + 5.49T + 47T^{2} \)
53 \( 1 + (5.29 - 9.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.24 + 5.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.21 - 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.10 - 1.90i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.02 - 5.23i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.13T + 73T^{2} \)
79 \( 1 + (-0.791 - 1.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.36 + 2.37i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.18 + 5.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.66T + 97T^{2} \)
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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

−8.905559657294241063623399032219, −8.014386246322643948393759354612, −7.23165167423244578377161425711, −6.67996348978805771917003605801, −5.72323838263027155144574289688, −5.21589817833750310529755204800, −3.78106883127715385308417874957, −2.46240609927516294797380713454, −1.61293053461329072391077632063, −0.01984556041413666235971893209, 1.77039513022241383330238562517, 3.37876822576965473367858242746, 4.14101386350167419007452015365, 5.15747248152487091026740373403, 5.54506222596569207107811155039, 6.49983587288265621460303463458, 7.87079500484033121541126374319, 8.282439211053898147033074764665, 9.489082685700940548125818738712, 10.08549470291391191043635898143

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