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Properties

Label 2-756-63.20-c1-0-0
Degree $2$
Conductor $756$
Sign $-0.899 - 0.435i$
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  + (0.276 + 0.479i)5-s + (−0.519 + 2.59i)7-s + (−4.03 − 2.32i)11-s + (−3.58 + 2.06i)13-s − 7.24·17-s + 6.71i·19-s + (4.85 − 2.80i)23-s + (2.34 − 4.06i)25-s + (−1.16 − 0.673i)29-s + (0.830 − 0.479i)31-s + (−1.38 + 0.469i)35-s − 7.06·37-s + (−2.39 − 4.14i)41-s + (−1.02 + 1.78i)43-s + (−4.90 + 8.49i)47-s + ⋯
L(s)  = 1  + (0.123 + 0.214i)5-s + (−0.196 + 0.980i)7-s + (−1.21 − 0.702i)11-s + (−0.993 + 0.573i)13-s − 1.75·17-s + 1.54i·19-s + (1.01 − 0.584i)23-s + (0.469 − 0.812i)25-s + (−0.216 − 0.125i)29-s + (0.149 − 0.0861i)31-s + (−0.234 + 0.0793i)35-s − 1.16·37-s + (−0.373 − 0.646i)41-s + (−0.156 + 0.271i)43-s + (−0.715 + 1.23i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.899 - 0.435i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.899 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.118262 + 0.515421i\)
\(L(\frac12)\) \(\approx\) \(0.118262 + 0.515421i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.519 - 2.59i)T \)
good5 \( 1 + (-0.276 - 0.479i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.03 + 2.32i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.58 - 2.06i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.24T + 17T^{2} \)
19 \( 1 - 6.71iT - 19T^{2} \)
23 \( 1 + (-4.85 + 2.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.16 + 0.673i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.830 + 0.479i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.06T + 37T^{2} \)
41 \( 1 + (2.39 + 4.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.02 - 1.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.90 - 8.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.43iT - 53T^{2} \)
59 \( 1 + (-3.89 - 6.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.37 - 3.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.68 - 2.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.407iT - 71T^{2} \)
73 \( 1 - 8.63iT - 73T^{2} \)
79 \( 1 + (0.318 - 0.551i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.78 + 4.82i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.93T + 89T^{2} \)
97 \( 1 + (7.48 + 4.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.66488206893358954685329399808, −9.919207595463309887209245868827, −8.861538055549541002156854592159, −8.352961322245626375061275041354, −7.17555563491803607874036963287, −6.28646813986403860207351763312, −5.40054998431229902896687461448, −4.46119275355101331798860553035, −2.93765353149853403631705875490, −2.17771206701972750982588066524, 0.24391357367376448691699942165, 2.16085933784731287214871894513, 3.30794882235193021704130171202, 4.88588936225350537621009661121, 5.01464179985928938875663866684, 6.95083051543628034529386048408, 7.04708589107146916608315873700, 8.240908565124044587980623766424, 9.232638256066012667919862192152, 10.00639898589215604487316103931

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