Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $0.514 + 0.857i$
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 + (−0.0627 − 2.64i)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 + (−0.0237 − 0.999i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.514 + 0.857i$
motivic weight  =  \(1\)
character  :  $\chi_{315} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 315,\ (\ :1/2),\ 0.514 + 0.857i)\)
\(L(1)\)  \(\approx\)  \(1.33014 - 0.753584i\)
\(L(\frac12)\)  \(\approx\)  \(1.33014 - 0.753584i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.0627 + 2.64i)T \)
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

−11.11430480603223487020957939701, −10.45150536827679485716922638970, −9.918098676036430409574484837898, −8.872821121753947249727714014126, −7.42769271432279266592372721067, −6.57898565367512422038512666364, −5.44334496287685179762505155303, −4.74141361877494719183082978021, −2.84229802130577099780774660636, −1.19096033079871757910733091631, 2.27287891502411944952842117957, 3.13854169663514469354244229471, 4.95547901902850622412064086815, 5.67778346620472574672022229179, 7.09639020915912047236977737502, 8.011948381746096294063148664699, 9.131349110677131913016762042813, 9.717401691808611746069009622809, 11.10424435705337747218800462319, 11.82511960056131955921051450066

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