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Properties

Label 2-1456-13.10-c1-0-6
Degree $2$
Conductor $1456$
Sign $-0.933 + 0.358i$
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  + (1.56 + 2.71i)3-s + 2.28i·5-s + (−0.866 − 0.5i)7-s + (−3.40 + 5.90i)9-s + (−4.77 + 2.75i)11-s + (−2.83 − 2.23i)13-s + (−6.19 + 3.57i)15-s + (2.80 − 4.85i)17-s + (4.25 + 2.45i)19-s − 3.13i·21-s + (0.971 + 1.68i)23-s − 0.208·25-s − 11.9·27-s + (−1.89 − 3.28i)29-s + 0.689i·31-s + ⋯
L(s)  = 1  + (0.904 + 1.56i)3-s + 1.02i·5-s + (−0.327 − 0.188i)7-s + (−1.13 + 1.96i)9-s + (−1.44 + 0.831i)11-s + (−0.785 − 0.618i)13-s + (−1.59 + 0.923i)15-s + (0.680 − 1.17i)17-s + (0.977 + 0.564i)19-s − 0.683i·21-s + (0.202 + 0.350i)23-s − 0.0417·25-s − 2.30·27-s + (−0.352 − 0.610i)29-s + 0.123i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.933 + 0.358i$
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: \((2,\ 1456,\ (\ :1/2),\ -0.933 + 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.437887498\)
\(L(\frac12)\) \(\approx\) \(1.437887498\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.83 + 2.23i)T \)
good3 \( 1 + (-1.56 - 2.71i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.28iT - 5T^{2} \)
11 \( 1 + (4.77 - 2.75i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.80 + 4.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.25 - 2.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.971 - 1.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.89 + 3.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.689iT - 31T^{2} \)
37 \( 1 + (5.40 - 3.12i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.61 - 2.08i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.11 - 7.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.55iT - 47T^{2} \)
53 \( 1 - 6.06T + 53T^{2} \)
59 \( 1 + (-12.9 - 7.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.70 + 6.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.90 - 4.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.7 - 6.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.00iT - 73T^{2} \)
79 \( 1 + 6.90T + 79T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 + (7.21 - 4.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.58 - 2.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.987662296806756929367316122156, −9.610584151032255929584175153691, −8.354723720166351558260582233500, −7.62452634698528330592550053955, −7.04217876149058743534215452712, −5.31095460077565278399718972616, −5.09906567438423396626251737568, −3.79485900797273977572986247119, −2.96271416538709195801014014597, −2.55619717596315010379609872709, 0.48187673897634962045157984059, 1.68594966243228961234536310098, 2.67683229709917115784169099818, 3.55094653631365463909641398115, 5.12426814904163254555468738902, 5.76450001368626580745894265637, 6.92217318796626112400512903267, 7.49136634691758208394712851419, 8.433352514237462229448125216125, 8.641249954198766683285162076501

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