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Properties

Label 2-756-84.23-c1-0-31
Degree $2$
Conductor $756$
Sign $0.170 - 0.985i$
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  + (−0.109 + 1.40i)2-s + (−1.97 − 0.307i)4-s + (3.44 + 1.99i)5-s + (0.212 − 2.63i)7-s + (0.649 − 2.75i)8-s + (−3.18 + 4.64i)10-s + (0.936 + 1.62i)11-s + 1.05·13-s + (3.69 + 0.587i)14-s + (3.81 + 1.21i)16-s + (2.30 − 1.33i)17-s + (0.628 + 0.362i)19-s + (−6.20 − 4.99i)20-s + (−2.38 + 1.14i)22-s + (3.00 − 5.20i)23-s + ⋯
L(s)  = 1  + (−0.0771 + 0.997i)2-s + (−0.988 − 0.153i)4-s + (1.54 + 0.890i)5-s + (0.0804 − 0.996i)7-s + (0.229 − 0.973i)8-s + (−1.00 + 1.46i)10-s + (0.282 + 0.489i)11-s + 0.293·13-s + (0.987 + 0.157i)14-s + (0.952 + 0.304i)16-s + (0.558 − 0.322i)17-s + (0.144 + 0.0831i)19-s + (−1.38 − 1.11i)20-s + (−0.509 + 0.243i)22-s + (0.626 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.170 - 0.985i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.170 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39930 + 1.17769i\)
\(L(\frac12)\) \(\approx\) \(1.39930 + 1.17769i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.109 - 1.40i)T \)
3 \( 1 \)
7 \( 1 + (-0.212 + 2.63i)T \)
good5 \( 1 + (-3.44 - 1.99i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.936 - 1.62i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.05T + 13T^{2} \)
17 \( 1 + (-2.30 + 1.33i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.628 - 0.362i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.00 + 5.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.09iT - 29T^{2} \)
31 \( 1 + (3.48 - 2.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.48 + 7.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.19iT - 41T^{2} \)
43 \( 1 - 12.3iT - 43T^{2} \)
47 \( 1 + (-2.03 + 3.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.2 - 6.48i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.56 - 7.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.21 - 9.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.00 - 2.88i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 + (7.37 + 12.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.92 + 2.84i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.27T + 83T^{2} \)
89 \( 1 + (-8.80 - 5.08i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.03T + 97T^{2} \)
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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

−10.44936515609579028263465133998, −9.564367956732565910599989569140, −8.988416928596944514911310867990, −7.60151112887899518407674124468, −6.99264690578156754579772318917, −6.27608000313181184963282256029, −5.43329501654238150454954495160, −4.36376057564028438810096396762, −3.04801596246433218108472441591, −1.37992524110029741813017851871, 1.25059542712530604939566680645, 2.16609323747609188204707127962, 3.34336085223376999006493928892, 4.81919576604562825746482461168, 5.54551849499490831987266376529, 6.19783716696381966844724678119, 8.055209745197265625416813671820, 8.766131791291302063406696672665, 9.522914134974167513505592801228, 9.847284483443191575795897085955

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