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Properties

Label 2-1456-13.4-c1-0-12
Degree $2$
Conductor $1456$
Sign $0.762 - 0.647i$
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  + (0.223 − 0.387i)3-s − 0.513i·5-s + (−0.866 + 0.5i)7-s + (1.39 + 2.42i)9-s + (1.05 + 0.606i)11-s + (−0.600 − 3.55i)13-s + (−0.199 − 0.115i)15-s + (3.32 + 5.76i)17-s + (−5.39 + 3.11i)19-s + 0.447i·21-s + (0.100 − 0.173i)23-s + 4.73·25-s + 2.59·27-s + (−0.904 + 1.56i)29-s + 0.945i·31-s + ⋯
L(s)  = 1  + (0.129 − 0.223i)3-s − 0.229i·5-s + (−0.327 + 0.188i)7-s + (0.466 + 0.808i)9-s + (0.316 + 0.182i)11-s + (−0.166 − 0.986i)13-s + (−0.0514 − 0.0296i)15-s + (0.807 + 1.39i)17-s + (−1.23 + 0.714i)19-s + 0.0977i·21-s + (0.0209 − 0.0362i)23-s + 0.947·25-s + 0.499·27-s + (−0.167 + 0.290i)29-s + 0.169i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.762 - 0.647i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.762 - 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.670850520\)
\(L(\frac12)\) \(\approx\) \(1.670850520\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.600 + 3.55i)T \)
good3 \( 1 + (-0.223 + 0.387i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.513iT - 5T^{2} \)
11 \( 1 + (-1.05 - 0.606i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.32 - 5.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.39 - 3.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.100 + 0.173i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.904 - 1.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.945iT - 31T^{2} \)
37 \( 1 + (-0.151 - 0.0873i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.45 - 3.14i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.13 + 1.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.38iT - 47T^{2} \)
53 \( 1 - 9.32T + 53T^{2} \)
59 \( 1 + (12.8 - 7.39i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.46 - 7.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.09 - 3.51i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.82 + 5.67i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.60iT - 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 0.866iT - 83T^{2} \)
89 \( 1 + (-6.67 - 3.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.3 - 7.70i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.674096925063876814869046649146, −8.602210457448659503116721406427, −8.096023797687915129223566995942, −7.28737150939698157029470045350, −6.29022812292064874809388646258, −5.54636479758673706908881539082, −4.53380308048875304824985732950, −3.58819771355891175881923651771, −2.40445895739250599277110609607, −1.28122494877889166652738820180, 0.73896740530563543122104561006, 2.33958665539969000651915157758, 3.42296261876935485129675375319, 4.25106316929777467126247227979, 5.14189245991706532446344130358, 6.50028477430778801854779704366, 6.77601655172173187996398085630, 7.73218942242908593887364165006, 8.952958624158323906421644604216, 9.324736350726828559198079780572

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