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Properties

Label 2-756-63.25-c1-0-5
Degree $2$
Conductor $756$
Sign $-0.0477 + 0.998i$
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 − 1.73i)5-s + (−2.5 − 0.866i)7-s + (2 + 3.46i)11-s + (−1.5 − 2.59i)13-s + (3.5 − 6.06i)17-s + (−2.5 − 4.33i)19-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + (−0.5 + 0.866i)29-s − 3·31-s + (−4 + 3.46i)35-s + (−5.5 − 9.52i)37-s + (−4.5 − 7.79i)41-s + (−2.5 + 4.33i)43-s − 3·47-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (−0.944 − 0.327i)7-s + (0.603 + 1.04i)11-s + (−0.416 − 0.720i)13-s + (0.848 − 1.47i)17-s + (−0.573 − 0.993i)19-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + (−0.0928 + 0.160i)29-s − 0.538·31-s + (−0.676 + 0.585i)35-s + (−0.904 − 1.56i)37-s + (−0.702 − 1.21i)41-s + (−0.381 + 0.660i)43-s − 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0477 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.0477 + 0.998i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.0477 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.891917 - 0.935586i\)
\(L(\frac12)\) \(\approx\) \(0.891917 - 0.935586i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 9T + 79T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.871751115675098161112653304834, −9.418319934230174856377597976382, −8.643162652326933815121367626502, −7.23101791822365787235798964611, −6.87302740285976810536360452389, −5.47343146164123109187481022185, −4.83102811244788963470027481560, −3.57689741879840005906462568462, −2.32836138723172585613220795480, −0.66143563828744786385086072334, 1.72881084394999007448461093778, 3.14894096657691983778769241616, 3.81509550361383763978718273989, 5.47038022724303532255153842673, 6.32589036732423119887338415682, 6.73584013999702334254314109400, 8.103468648819423980249647960001, 8.879464896301950656724252178718, 9.928632641684825750663932613682, 10.31313144760867943305963275787

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