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Properties

Label 2-1456-7.4-c1-0-39
Degree $2$
Conductor $1456$
Sign $-0.165 + 0.986i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + (1.23 − 2.14i)3-s + (1.06 + 1.83i)5-s + (−2.63 − 0.272i)7-s + (−1.56 − 2.70i)9-s + (2.39 − 4.14i)11-s + 13-s + 5.25·15-s + (1.88 − 3.27i)17-s + (−1.78 − 3.08i)19-s + (−3.83 + 5.30i)21-s + (2.23 + 3.87i)23-s + (0.246 − 0.427i)25-s − 0.303·27-s − 5.90·29-s + (−1.88 + 3.26i)31-s + ⋯
L(s)  = 1  + (0.714 − 1.23i)3-s + (0.474 + 0.822i)5-s + (−0.994 − 0.102i)7-s + (−0.520 − 0.901i)9-s + (0.721 − 1.25i)11-s + 0.277·13-s + 1.35·15-s + (0.458 − 0.793i)17-s + (−0.409 − 0.708i)19-s + (−0.837 + 1.15i)21-s + (0.466 + 0.807i)23-s + (0.0493 − 0.0855i)25-s − 0.0584·27-s − 1.09·29-s + (−0.338 + 0.586i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.165 + 0.986i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.077840967\)
\(L(\frac12)\) \(\approx\) \(2.077840967\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (2.63 + 0.272i)T \)
13 \( 1 - T \)
good3 \( 1 + (-1.23 + 2.14i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.06 - 1.83i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.39 + 4.14i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.88 + 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.78 + 3.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.90T + 29T^{2} \)
31 \( 1 + (1.88 - 3.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.81 + 4.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 3.40T + 43T^{2} \)
47 \( 1 + (3.55 + 6.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.19 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.39 + 4.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.60 + 2.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.44 - 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + (3.85 - 6.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.58 + 4.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + (1.83 + 3.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.40T + 97T^{2} \)
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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

−9.116032722970558226865563076139, −8.580674056123888321627245993912, −7.43097654908803567542669244032, −6.91675948551493455021855788519, −6.31749856519704235387285510368, −5.48773280249254808505146353800, −3.64264533303916724942465084664, −3.07180162983167566621298678053, −2.14819976081084681209895628922, −0.78386515899032747926816796072, 1.57417528781564671298395823363, 2.87400493448617184239428617983, 3.95464454188743556836929296556, 4.38711991095515347815732278219, 5.51754396783819158568287294390, 6.32798907204166739103025103924, 7.41582602060143716710102142701, 8.547565614519080563896973231063, 9.119475289585558120689010654270, 9.644033502489734630956479086616

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