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Properties

Label 2-756-21.5-c1-0-9
Degree $2$
Conductor $756$
Sign $-0.895 + 0.444i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (0.385 − 0.667i)5-s + (−2.20 + 1.46i)7-s + (−4.68 + 2.70i)11-s − 5.28i·13-s + (−2.85 − 4.94i)17-s + (0.535 + 0.309i)19-s + (−5.83 − 3.37i)23-s + (2.20 + 3.81i)25-s − 8.07i·29-s + (0.502 − 0.290i)31-s + (0.129 + 2.03i)35-s + (−4.53 + 7.86i)37-s − 8.59·41-s + 3.40·43-s + (0.385 − 0.667i)47-s + ⋯
L(s)  = 1  + (0.172 − 0.298i)5-s + (−0.832 + 0.553i)7-s + (−1.41 + 0.814i)11-s − 1.46i·13-s + (−0.692 − 1.19i)17-s + (0.122 + 0.0709i)19-s + (−1.21 − 0.702i)23-s + (0.440 + 0.763i)25-s − 1.49i·29-s + (0.0902 − 0.0521i)31-s + (0.0218 + 0.344i)35-s + (−0.746 + 1.29i)37-s − 1.34·41-s + 0.519·43-s + (0.0562 − 0.0973i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0780859 - 0.333328i\)
\(L(\frac12)\) \(\approx\) \(0.0780859 - 0.333328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (2.20 - 1.46i)T \)
good5 \( 1 + (-0.385 + 0.667i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.68 - 2.70i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.28iT - 13T^{2} \)
17 \( 1 + (2.85 + 4.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.535 - 0.309i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.83 + 3.37i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.07iT - 29T^{2} \)
31 \( 1 + (-0.502 + 0.290i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.53 - 7.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.59T + 41T^{2} \)
43 \( 1 - 3.40T + 43T^{2} \)
47 \( 1 + (-0.385 + 0.667i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.63 - 3.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.89 - 11.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.57 + 2.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.77 + 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.25iT - 71T^{2} \)
73 \( 1 + (-1.96 + 1.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.03 + 6.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.85T + 83T^{2} \)
89 \( 1 + (-2.59 + 4.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.0iT - 97T^{2} \)
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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

−10.05807091119841965923569841832, −9.233229119391778244488980011620, −8.207928381083100783716538323128, −7.49460076126499852093446413081, −6.37452005231380049421652889744, −5.43438640681381670575602146560, −4.71625496355985467840121586742, −3.13198235629866008590152670894, −2.33133079289665050853339521607, −0.15856062597472183592407514565, 1.97200994478841536384863448490, 3.26028337373767865831689103333, 4.18767963520195615375072260114, 5.47587736736022089514632680894, 6.42292749228436096238805556531, 7.08638234319422461221174707947, 8.203576115252033017944527130696, 8.981630449226691662173799004270, 10.03474416875388187977273370425, 10.60788392199137192566493647690

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