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Properties

Label 2-756-189.173-c1-0-2
Degree $2$
Conductor $756$
Sign $-0.950 - 0.309i$
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  + (−1.16 + 1.27i)3-s + (0.898 − 0.753i)5-s + (−1.31 − 2.29i)7-s + (−0.267 − 2.98i)9-s + (−2.05 + 2.45i)11-s + (1.12 + 1.33i)13-s + (−0.0867 + 2.02i)15-s + (−1.58 + 2.74i)17-s + (−5.76 + 3.32i)19-s + (4.47 + 0.997i)21-s + (−1.94 + 5.34i)23-s + (−0.629 + 3.56i)25-s + (4.13 + 3.15i)27-s + (3.84 − 4.58i)29-s + (−2.86 − 3.41i)31-s + ⋯
L(s)  = 1  + (−0.674 + 0.737i)3-s + (0.401 − 0.337i)5-s + (−0.498 − 0.867i)7-s + (−0.0890 − 0.996i)9-s + (−0.620 + 0.739i)11-s + (0.311 + 0.371i)13-s + (−0.0223 + 0.524i)15-s + (−0.383 + 0.664i)17-s + (−1.32 + 0.763i)19-s + (0.976 + 0.217i)21-s + (−0.405 + 1.11i)23-s + (−0.125 + 0.713i)25-s + (0.795 + 0.606i)27-s + (0.714 − 0.851i)29-s + (−0.514 − 0.612i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.950 - 0.309i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.950 - 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0583979 + 0.367534i\)
\(L(\frac12)\) \(\approx\) \(0.0583979 + 0.367534i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 - 1.27i)T \)
7 \( 1 + (1.31 + 2.29i)T \)
good5 \( 1 + (-0.898 + 0.753i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (2.05 - 2.45i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.12 - 1.33i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.58 - 2.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.76 - 3.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.94 - 5.34i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-3.84 + 4.58i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.86 + 3.41i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + 9.50T + 37T^{2} \)
41 \( 1 + (7.74 - 6.50i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.80 + 1.38i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (6.48 + 5.44i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-11.7 + 6.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.90 - 10.7i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.66 + 1.98i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (14.4 + 5.25i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-2.08 + 1.20i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.16iT - 73T^{2} \)
79 \( 1 + (9.42 - 3.43i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-5.29 - 4.44i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-5.74 - 9.95i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.0688 - 0.189i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.37184035231106047544306957235, −10.17214042361814710769513974267, −9.256673136097833386714997429645, −8.264359679065482211512229879101, −7.06688554006874886135080532093, −6.24403074610601637930970710096, −5.36778142188733466801531118070, −4.31169756397784126290560929298, −3.62553111173389522136855193046, −1.76385161095677844703426698171, 0.19320445863608968163833612452, 2.14560940111597181122359248354, 2.98957539209261960878142674499, 4.78434610465111681440422966199, 5.68565205470378019225089719803, 6.42037772626222071327752830518, 7.04938686684068835847787793788, 8.450546254385770433475089729138, 8.795574266162310537894490747896, 10.40346626578271850047661482636

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