Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $-0.881 + 0.472i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.167 + 0.167i)2-s − 1.94i·4-s + (−2.23 − 0.0836i)5-s + (−2.64 − 0.0627i)7-s + (0.658 − 0.658i)8-s + (−0.359 − 0.387i)10-s − 3.98·11-s + (0.500 + 0.500i)13-s + (−0.431 − 0.452i)14-s − 3.66·16-s + (1.67 − 1.67i)17-s − 7.21·19-s + (−0.162 + 4.34i)20-s + (−0.665 − 0.665i)22-s + (5.16 − 5.16i)23-s + ⋯
L(s)  = 1  + (0.118 + 0.118i)2-s − 0.972i·4-s + (−0.999 − 0.0373i)5-s + (−0.999 − 0.0237i)7-s + (0.232 − 0.232i)8-s + (−0.113 − 0.122i)10-s − 1.20·11-s + (0.138 + 0.138i)13-s + (−0.115 − 0.120i)14-s − 0.917·16-s + (0.407 − 0.407i)17-s − 1.65·19-s + (−0.0363 + 0.971i)20-s + (−0.141 − 0.141i)22-s + (1.07 − 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.881 + 0.472i$
motivic weight  =  \(1\)
character  :  $\chi_{315} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 315,\ (\ :1/2),\ -0.881 + 0.472i)\)
\(L(1)\)  \(\approx\)  \(0.115786 - 0.460685i\)
\(L(\frac12)\)  \(\approx\)  \(0.115786 - 0.460685i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.23 + 0.0836i)T \)
7 \( 1 + (2.64 + 0.0627i)T \)
good2 \( 1 + (-0.167 - 0.167i)T + 2iT^{2} \)
11 \( 1 + 3.98T + 11T^{2} \)
13 \( 1 + (-0.500 - 0.500i)T + 13iT^{2} \)
17 \( 1 + (-1.67 + 1.67i)T - 17iT^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 + (-5.16 + 5.16i)T - 23iT^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + 4.93iT - 31T^{2} \)
37 \( 1 + (-0.292 - 0.292i)T + 37iT^{2} \)
41 \( 1 + 7.63iT - 41T^{2} \)
43 \( 1 + (-3.65 + 3.65i)T - 43iT^{2} \)
47 \( 1 + (-0.305 + 0.305i)T - 47iT^{2} \)
53 \( 1 + (5.39 - 5.39i)T - 53iT^{2} \)
59 \( 1 + 6.10T + 59T^{2} \)
61 \( 1 + 7.11iT - 61T^{2} \)
67 \( 1 + (-0.944 - 0.944i)T + 67iT^{2} \)
71 \( 1 + 1.19T + 71T^{2} \)
73 \( 1 + (-1.38 - 1.38i)T + 73iT^{2} \)
79 \( 1 - 8.64iT - 79T^{2} \)
83 \( 1 + (-11.9 - 11.9i)T + 83iT^{2} \)
89 \( 1 + 7.82T + 89T^{2} \)
97 \( 1 + (-7.43 + 7.43i)T - 97iT^{2} \)
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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

−10.88311861963119136469130974631, −10.63122388481666987365757210403, −9.427168889616118089781855525232, −8.466947032021894115404545232007, −7.24052367170170262144812931467, −6.40741713608982723237069848590, −5.22813011601867060406011859817, −4.15057511818560801211776362846, −2.66770414275949976815382019912, −0.31264738796172446026474888106, 2.79652787539199953842597958966, 3.63325013916053785577592202895, 4.78308697944975154370870541449, 6.35547210235828066837221591963, 7.45072455521196501036083586418, 8.131801333460831383403228660226, 9.061678493785444426362184236070, 10.41167322278117803782152212393, 11.17706986619612525031981754813, 12.20778353530963949587965272068

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