Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 7 $
Sign $0.999 + 0.0348i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)2-s + (−1.04 − 1.38i)3-s + (0.766 − 0.642i)4-s + (0.108 + 0.617i)5-s + (1.45 + 0.940i)6-s + (0.766 + 0.642i)7-s + (−0.500 + 0.866i)8-s + (−0.815 + 2.88i)9-s + (−0.313 − 0.542i)10-s + (−0.0434 + 0.246i)11-s + (−1.68 − 0.386i)12-s + (3.98 + 1.45i)13-s + (−0.939 − 0.342i)14-s + (0.739 − 0.795i)15-s + (0.173 − 0.984i)16-s + (0.0870 + 0.150i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−0.603 − 0.797i)3-s + (0.383 − 0.321i)4-s + (0.0486 + 0.276i)5-s + (0.593 + 0.383i)6-s + (0.289 + 0.242i)7-s + (−0.176 + 0.306i)8-s + (−0.271 + 0.962i)9-s + (−0.0991 − 0.171i)10-s + (−0.0130 + 0.0742i)11-s + (−0.487 − 0.111i)12-s + (1.10 + 0.402i)13-s + (−0.251 − 0.0914i)14-s + (0.190 − 0.205i)15-s + (0.0434 − 0.246i)16-s + (0.0211 + 0.0365i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.999 + 0.0348i$
motivic weight  =  \(1\)
character  :  $\chi_{378} (169, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 378,\ (\ :1/2),\ 0.999 + 0.0348i)$
$L(1)$  $\approx$  $0.880191 - 0.0153432i$
$L(\frac12)$  $\approx$  $0.880191 - 0.0153432i$
$L(\frac{3}{2})$   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 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (1.04 + 1.38i)T \)
7 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (-0.108 - 0.617i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (0.0434 - 0.246i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-3.98 - 1.45i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.0870 - 0.150i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.22 + 2.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0380 + 0.0319i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.635 + 0.231i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-5.73 + 4.81i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-2.59 - 4.50i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-11.3 - 4.14i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.237 + 1.34i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.339 - 0.285i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + (-0.847 - 4.80i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.13 + 1.79i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.13 + 1.14i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (2.14 + 3.71i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.93 - 8.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.12 + 1.86i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-5.81 + 2.11i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-3.53 + 6.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.64 - 15.0i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.25178426981754904521664898966, −10.66217597536045543661719579616, −9.440779201808952120950726703022, −8.410413720682314146068328117917, −7.64423717478983407362697057741, −6.56503564335252175970044995582, −5.98365847536767597251956336158, −4.65915801822237368572504789076, −2.64712349851005546416659957992, −1.17125434997677926028711906738, 1.06425874539303324001085827736, 3.18837348496177616236648340716, 4.35331110467483203635562980464, 5.55877607665463289332495634299, 6.53831832277671753809657767269, 7.86625236768918847858880158124, 8.808030031962588574214448910512, 9.577257999303747050888410160961, 10.65652509160657801015357415883, 10.98167375635548391608161451491

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