Properties

Degree 2
Conductor $ 2^{2} \cdot 37 $
Sign $0.763 - 0.646i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − 7-s i·11-s + (−1 − i)17-s + (1 + i)19-s i·21-s + (−1 − i)23-s i·25-s + i·27-s + (−1 + i)29-s + 33-s + i·37-s + i·41-s + 47-s + (1 − i)51-s + ⋯
L(s)  = 1  + i·3-s − 7-s i·11-s + (−1 − i)17-s + (1 + i)19-s i·21-s + (−1 − i)23-s i·25-s + i·27-s + (−1 + i)29-s + 33-s + i·37-s + i·41-s + 47-s + (1 − i)51-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(148\)    =    \(2^{2} \cdot 37\)
\( \varepsilon \)  =  $0.763 - 0.646i$
motivic weight  =  \(0\)
character  :  $\chi_{148} (117, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 148,\ (\ :0),\ 0.763 - 0.646i)$
$L(\frac{1}{2})$  $\approx$  $0.6069956411$
$L(\frac12)$  $\approx$  $0.6069956411$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;37\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;37\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
37 \( 1 - iT \)
good3 \( 1 - iT - T^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + (1 - i)T - iT^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + (1 + i)T + iT^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (1 - i)T - iT^{2} \)
97 \( 1 + iT^{2} \)
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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

−13.49148832149289409247800683737, −12.38211825179779823233357558045, −11.23585659445774750887200017992, −10.19320403310866793307351058503, −9.522405592957802838890433163166, −8.444784212317855955232376186194, −6.89050957330215446206173448352, −5.69019913107289534205610998386, −4.27714347876596393041059830264, −3.09928043927083506759952653755, 2.08628502080588857118603629212, 3.95290976747288843243538632860, 5.77021853794458276231113710756, 6.94968605451004120504551546434, 7.56795461850848674019017603429, 9.141281680914141676207393897133, 9.990264392653704431458007765550, 11.37649815088232660667564705338, 12.43625305431146567598223854948, 13.10816457337357686041792780832

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