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Properties

Label 2-648-9.7-c1-0-2
Degree $2$
Conductor $648$
Sign $-0.342 - 0.939i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
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  + (1.86 + 3.23i)5-s + (−1.73 + 3i)7-s + (1 − 1.73i)11-s + (1.23 + 2.13i)13-s − 2.26·17-s − 7.46·19-s + (−2.46 − 4.26i)23-s + (−4.46 + 7.73i)25-s + (2.13 − 3.69i)29-s + (5.46 + 9.46i)31-s − 12.9·35-s − 0.464·37-s + (3.46 + 6i)41-s + (2.26 − 3.92i)43-s + (3.46 − 6i)47-s + ⋯
L(s)  = 1  + (0.834 + 1.44i)5-s + (−0.654 + 1.13i)7-s + (0.301 − 0.522i)11-s + (0.341 + 0.591i)13-s − 0.550·17-s − 1.71·19-s + (−0.513 − 0.889i)23-s + (−0.892 + 1.54i)25-s + (0.396 − 0.686i)29-s + (0.981 + 1.69i)31-s − 2.18·35-s − 0.0762·37-s + (0.541 + 0.937i)41-s + (0.345 − 0.599i)43-s + (0.505 − 0.875i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.803827 + 1.14798i\)
\(L(\frac12)\) \(\approx\) \(0.803827 + 1.14798i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-1.86 - 3.23i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.73 - 3i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.23 - 2.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 + 7.46T + 19T^{2} \)
23 \( 1 + (2.46 + 4.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.13 + 3.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.46 - 9.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.464T + 37T^{2} \)
41 \( 1 + (-3.46 - 6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.26 + 3.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.46 + 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.23 - 9.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.267 - 0.464i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (0.267 - 0.464i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.46 - 2.53i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + (-5.92 + 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60077381552566648312639035669, −10.17230732993925358068589104096, −9.005695093263614897205060637678, −8.510210156719578903667854725124, −6.84365271564091683279906536926, −6.39399928466460444669654418554, −5.79018973126271885165810825743, −4.18039995764769126180089180976, −2.84266783318905781118572179705, −2.20503263524625572612342910275, 0.74472293952866474962420748327, 2.11015797825674241479677131512, 3.92859583736378663934600248803, 4.60650802936672040788348898344, 5.81792926715268055459196533350, 6.56914772412111454538092192586, 7.75564102954044241309589187220, 8.677106304019078570860908699145, 9.483037619524655563714656114900, 10.13179255602022547138628758357

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