Properties

Degree $2$
Conductor $92778$
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 + 2·5-s − 6-s − 7-s + 8-s + 9-s + 2·10-s + 4·11-s − 12-s − 6·13-s − 14-s − 2·15-s + 16-s + 2·17-s + 18-s + 4·19-s + 2·20-s + 21-s + 4·22-s − 8·23-s − 24-s − 25-s − 6·26-s − 27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-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 − 0.267·14-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 + 0.218·21-s + 0.852·22-s − 1.66·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92778 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92778 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92778\)    =    \(2 \cdot 3 \cdot 7 \cdot 47^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{92778} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 92778,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 + T \)
47 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.02464299334945, −13.83248451647171, −13.01951659822444, −12.50773717237814, −12.13047378917093, −11.80804943848531, −11.39505280844051, −10.47098300331857, −10.12401692913173, −9.741181553975858, −9.362351062878588, −8.665427433809357, −7.812466344074141, −7.347168106133333, −6.823114730118541, −6.393749211897422, −5.634661791728373, −5.583491625811294, −4.879692401643342, −4.143095342946685, −3.800666547929241, −2.944073798444713, −2.317639243554027, −1.750683931081212, −1.022118967082164, 0, 1.022118967082164, 1.750683931081212, 2.317639243554027, 2.944073798444713, 3.800666547929241, 4.143095342946685, 4.879692401643342, 5.583491625811294, 5.634661791728373, 6.393749211897422, 6.823114730118541, 7.347168106133333, 7.812466344074141, 8.665427433809357, 9.362351062878588, 9.741181553975858, 10.12401692913173, 10.47098300331857, 11.39505280844051, 11.80804943848531, 12.13047378917093, 12.50773717237814, 13.01951659822444, 13.83248451647171, 14.02464299334945

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