Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 29 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s − 4·11-s + 12-s + 6·13-s + 2·15-s + 16-s − 2·17-s − 18-s + 4·19-s + 2·20-s + 4·22-s − 24-s − 25-s − 6·26-s + 27-s − 29-s − 2·30-s − 4·31-s − 32-s − 4·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s + 1.66·13-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.185·29-s − 0.365·30-s − 0.718·31-s − 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(174\)    =    \(2 \cdot 3 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{174} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 174,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.138222379$
$L(\frac12)$  $\approx$  $1.138222379$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
29 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−19.88508407481155, −18.78412790480485, −18.13350868150038, −17.62264267140367, −16.18184242153092, −15.81197537687345, −14.60520209287848, −13.47484752443243, −13.08350822284661, −11.46160956762346, −10.51228751975808, −9.664379188026696, −8.703567572426078, −7.834198620535603, −6.519626517228254, −5.355230447865983, −3.325129021266584, −1.819771036615822, 1.819771036615822, 3.325129021266584, 5.355230447865983, 6.519626517228254, 7.834198620535603, 8.703567572426078, 9.664379188026696, 10.51228751975808, 11.46160956762346, 13.08350822284661, 13.47484752443243, 14.60520209287848, 15.81197537687345, 16.18184242153092, 17.62264267140367, 18.13350868150038, 18.78412790480485, 19.88508407481155

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