Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.644 + 0.764i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.487 − 1.66i)3-s + (2.74 − 4.27i)7-s + (−2.52 − 1.62i)9-s + (−6.83 + 0.983i)13-s + (7.27 − 4.67i)19-s + (−5.75 − 6.64i)21-s + (0.711 + 4.94i)25-s + (−3.92 + 3.40i)27-s + (−9.72 − 1.39i)31-s + 5.43·37-s + (−1.70 + 11.8i)39-s + (5.85 + 2.67i)43-s + (−7.80 − 17.0i)49-s + (−4.22 − 14.3i)57-s + (8.00 − 6.93i)61-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)3-s + (1.03 − 1.61i)7-s + (−0.841 − 0.540i)9-s + (−1.89 + 0.272i)13-s + (1.66 − 1.07i)19-s + (−1.25 − 1.45i)21-s + (0.142 + 0.989i)25-s + (−0.755 + 0.654i)27-s + (−1.74 − 0.251i)31-s + 0.893·37-s + (−0.272 + 1.89i)39-s + (0.893 + 0.407i)43-s + (−1.11 − 2.44i)49-s + (−0.559 − 1.90i)57-s + (1.02 − 0.888i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.644 + 0.764i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (125, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.644 + 0.764i)$
$L(1)$  $\approx$  $0.657917 - 1.41433i$
$L(\frac12)$  $\approx$  $0.657917 - 1.41433i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;67\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.487 + 1.66i)T \)
67 \( 1 + (2.85 + 7.67i)T \)
good5 \( 1 + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (-2.74 + 4.27i)T + (-2.90 - 6.36i)T^{2} \)
11 \( 1 + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (6.83 - 0.983i)T + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (11.1 + 12.8i)T^{2} \)
19 \( 1 + (-7.27 + 4.67i)T + (7.89 - 17.2i)T^{2} \)
23 \( 1 + (-19.3 - 12.4i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (9.72 + 1.39i)T + (29.7 + 8.73i)T^{2} \)
37 \( 1 - 5.43T + 37T^{2} \)
41 \( 1 + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-5.85 - 2.67i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (-39.5 - 25.4i)T^{2} \)
53 \( 1 + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (56.6 + 16.6i)T^{2} \)
61 \( 1 + (-8.00 + 6.93i)T + (8.68 - 60.3i)T^{2} \)
71 \( 1 + (46.4 - 53.6i)T^{2} \)
73 \( 1 + (2.56 + 2.96i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-1.47 + 0.212i)T + (75.7 - 22.2i)T^{2} \)
83 \( 1 + (11.8 + 82.1i)T^{2} \)
89 \( 1 + (-74.8 + 48.1i)T^{2} \)
97 \( 1 - 9.35iT - 97T^{2} \)
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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

−9.819277701778313436712383570295, −9.160900600673507065623952033287, −7.79745347175948861194967673052, −7.39892750193068018150224004407, −6.98416574494615738299659834129, −5.39811109844985612821587229986, −4.59879289353777112347601769415, −3.30121876010478972578301820324, −1.99007463274771737683916455701, −0.74726171647869634602790386046, 2.13722620084439560032136645816, 2.95821328760056827486463075244, 4.38163994267355175955845127205, 5.36205155900616923923424142946, 5.61738742055562249544862754026, 7.44632133251895351493869431440, 8.102002330227842864001246163141, 9.023799200781772161579319235918, 9.623188190629059432361159343238, 10.40302339729587553186931136758

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