Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s − 2·7-s + 9-s − 4·11-s + 13-s + 4·15-s + 2·17-s − 2·19-s + 2·21-s + 11·25-s − 27-s − 6·29-s − 10·31-s + 4·33-s + 8·35-s + 10·37-s − 39-s + 8·41-s + 4·43-s − 4·45-s − 4·47-s − 3·49-s − 2·51-s − 10·53-s + 16·55-s + 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.03·15-s + 0.485·17-s − 0.458·19-s + 0.436·21-s + 11/5·25-s − 0.192·27-s − 1.11·29-s − 1.79·31-s + 0.696·33-s + 1.35·35-s + 1.64·37-s − 0.160·39-s + 1.24·41-s + 0.609·43-s − 0.596·45-s − 0.583·47-s − 3/7·49-s − 0.280·51-s − 1.37·53-s + 2.15·55-s + 0.264·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{156} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 156,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−19.66936246870811, −18.76856110398365, −18.29920029462316, −16.76327390589409, −16.13788626546098, −15.54440170287951, −14.63311062184908, −12.88254835221111, −12.60735281309984, −11.28731824529633, −10.81735681549711, −9.399166260213490, −7.965645572263158, −7.337869793459183, −5.892490652544999, −4.459843089107868, −3.272140766593661, 0, 3.272140766593661, 4.459843089107868, 5.892490652544999, 7.337869793459183, 7.965645572263158, 9.399166260213490, 10.81735681549711, 11.28731824529633, 12.60735281309984, 12.88254835221111, 14.63311062184908, 15.54440170287951, 16.13788626546098, 16.76327390589409, 18.29920029462316, 18.76856110398365, 19.66936246870811

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