Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 5·11-s − 12-s + 13-s + 14-s + 16-s + 17-s − 18-s + 21-s + 5·22-s − 6·23-s + 24-s − 5·25-s − 26-s − 27-s − 28-s − 10·29-s − 8·31-s − 32-s + 5·33-s − 34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s − 1.25·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s − 1.85·29-s − 1.43·31-s − 0.176·32-s + 0.870·33-s − 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(354\)    =    \(2 \cdot 3 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{354} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 354,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;59\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
59 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−10.91606193656124739003864976887, −10.09232308996604778849825434589, −9.363551266314125453468629303075, −8.025407693218525427980316031873, −7.43208854303745278135460909362, −6.10761877659726684085908706448, −5.37002895885674566259205833480, −3.72832421421558189975884042631, −2.12052920002859578562836029663, 0, 2.12052920002859578562836029663, 3.72832421421558189975884042631, 5.37002895885674566259205833480, 6.10761877659726684085908706448, 7.43208854303745278135460909362, 8.025407693218525427980316031873, 9.363551266314125453468629303075, 10.09232308996604778849825434589, 10.91606193656124739003864976887

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