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Properties

Label 2-1456-13.10-c1-0-15
Degree $2$
Conductor $1456$
Sign $0.129 - 0.991i$
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  + (1.19 + 2.06i)3-s − 0.877i·5-s + (0.866 + 0.5i)7-s + (−1.34 + 2.33i)9-s + (2.75 − 1.59i)11-s + (−3.31 + 1.42i)13-s + (1.81 − 1.04i)15-s + (−0.485 + 0.841i)17-s + (1.37 + 0.792i)19-s + 2.38i·21-s + (1.63 + 2.82i)23-s + 4.22·25-s + 0.733·27-s + (4.86 + 8.42i)29-s + 3.94i·31-s + ⋯
L(s)  = 1  + (0.688 + 1.19i)3-s − 0.392i·5-s + (0.327 + 0.188i)7-s + (−0.448 + 0.777i)9-s + (0.831 − 0.480i)11-s + (−0.918 + 0.395i)13-s + (0.468 − 0.270i)15-s + (−0.117 + 0.204i)17-s + (0.314 + 0.181i)19-s + 0.520i·21-s + (0.340 + 0.589i)23-s + 0.845·25-s + 0.141·27-s + (0.903 + 1.56i)29-s + 0.707i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.129 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.248668392\)
\(L(\frac12)\) \(\approx\) \(2.248668392\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (3.31 - 1.42i)T \)
good3 \( 1 + (-1.19 - 2.06i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.877iT - 5T^{2} \)
11 \( 1 + (-2.75 + 1.59i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.485 - 0.841i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.37 - 0.792i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.63 - 2.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.86 - 8.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.94iT - 31T^{2} \)
37 \( 1 + (4.47 - 2.58i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.91 - 2.25i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.81 + 6.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.12iT - 47T^{2} \)
53 \( 1 + 2.01T + 53T^{2} \)
59 \( 1 + (0.901 + 0.520i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.50 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.45 + 3.72i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.53 + 2.03i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 - 9.79T + 79T^{2} \)
83 \( 1 + 6.81iT - 83T^{2} \)
89 \( 1 + (-4.01 + 2.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.55 + 2.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.535519167564418450553571556465, −8.881157945592683349533974723018, −8.562065848664455796752381914868, −7.36237395601826975368972014020, −6.46620851571831911330756405081, −5.11963249174606590788368569233, −4.73966118997210499100382888831, −3.65931760714631073427648040770, −2.94044213462890448961232836037, −1.45055637644269810431933422056, 0.914485079059268755720724492913, 2.17988793211355973206362634055, 2.84924818748322728925397130022, 4.15861560442538718404138533981, 5.15102679828964744778346935496, 6.49708011925258269438070229385, 6.93721896385722158158834919521, 7.69807966735682258150385064450, 8.311434782487533273788087798905, 9.242054566758412983247620695217

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