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Properties

Label 2-1456-7.2-c1-0-17
Degree $2$
Conductor $1456$
Sign $-0.0489 + 0.998i$
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  + (−1.40 − 2.42i)3-s + (−1.92 + 3.32i)5-s + (−2.62 − 0.296i)7-s + (−2.42 + 4.19i)9-s + (0.228 + 0.396i)11-s + 13-s + 10.7·15-s + (1.02 + 1.76i)17-s + (0.708 − 1.22i)19-s + (2.96 + 6.79i)21-s + (−3.44 + 5.96i)23-s + (−4.87 − 8.45i)25-s + 5.15·27-s + 0.199·29-s + (−0.400 − 0.693i)31-s + ⋯
L(s)  = 1  + (−0.808 − 1.40i)3-s + (−0.858 + 1.48i)5-s + (−0.993 − 0.112i)7-s + (−0.806 + 1.39i)9-s + (0.0690 + 0.119i)11-s + 0.277·13-s + 2.77·15-s + (0.247 + 0.428i)17-s + (0.162 − 0.281i)19-s + (0.646 + 1.48i)21-s + (−0.717 + 1.24i)23-s + (−0.975 − 1.69i)25-s + 0.992·27-s + 0.0370·29-s + (−0.0718 − 0.124i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0489 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0489 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.0489 + 0.998i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.0489 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5370405519\)
\(L(\frac12)\) \(\approx\) \(0.5370405519\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.296i)T \)
13 \( 1 - T \)
good3 \( 1 + (1.40 + 2.42i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.92 - 3.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.228 - 0.396i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.02 - 1.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.708 + 1.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.44 - 5.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.199T + 29T^{2} \)
31 \( 1 + (0.400 + 0.693i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.85 + 6.68i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.95T + 41T^{2} \)
43 \( 1 - 8.64T + 43T^{2} \)
47 \( 1 + (-1.22 + 2.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.157 - 0.273i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.45 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.65 + 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.94 + 6.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + (7.17 + 12.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.92 - 8.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 + (7.69 - 13.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.80T + 97T^{2} \)
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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

−9.422809933075032888984616919479, −8.003100630055977578439966804522, −7.51466794768858033363712973699, −6.83454366353731856741465744562, −6.31539326586343347717699768376, −5.59652313078000483951920122508, −3.92657157875531704731326907773, −3.14907908660165076592717962986, −1.99279207489948746188162265146, −0.35086389330200191846204242938, 0.810044716544009018176718189920, 3.08199900698690273001005139040, 4.16956341303097414170701679077, 4.45125976944862284072563892415, 5.51156132497145005382228256864, 6.06476103087821499476242847336, 7.32723450758029338014514042470, 8.535927371152860308610403984022, 8.906855133063064217566684415786, 9.866366027397959362524768550647

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