Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7^{2} $
Sign $-0.386 - 0.922i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (2.5 + 4.33i)11-s + 0.999·15-s + (−2 − 3.46i)17-s + (−4 + 6.92i)19-s + (−2 + 3.46i)23-s + (2 + 3.46i)25-s − 0.999·27-s − 5·29-s + (−1.5 − 2.59i)31-s + (−2.5 + 4.33i)33-s + (2 − 3.46i)37-s − 2·43-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.166 + 0.288i)9-s + (0.753 + 1.30i)11-s + 0.258·15-s + (−0.485 − 0.840i)17-s + (−0.917 + 1.58i)19-s + (−0.417 + 0.722i)23-s + (0.400 + 0.692i)25-s − 0.192·27-s − 0.928·29-s + (−0.269 − 0.466i)31-s + (−0.435 + 0.753i)33-s + (0.328 − 0.569i)37-s − 0.304·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.386 - 0.922i$
motivic weight  =  \(1\)
character  :  $\chi_{2352} (961, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2352,\ (\ :1/2),\ -0.386 - 0.922i)\)
\(L(1)\)  \(\approx\)  \(1.583094808\)
\(L(\frac12)\)  \(\approx\)  \(1.583094808\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.264099594723781224880489550601, −8.674904681914837763667206653583, −7.62889102428376101164693166654, −7.07484583946968114303913920671, −5.97351072899392792649825708485, −5.25456993932562790526417908683, −4.24147627282244726410142343926, −3.80262749528093413368229290515, −2.38084421049568660477990229632, −1.53166094255051119392166571361, 0.49950331082421080325331240143, 1.89649472878622797180207758959, 2.81322368978783275942551703448, 3.73696130146704588161860501122, 4.67331392305461226897948667618, 5.90446912952814723134505483135, 6.49128971882048352704348200360, 6.98476033230961536257573073360, 8.210204336131024636060434124044, 8.623122995027749086326652796862

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