Properties

Label 2-379050-1.1-c1-0-95
Degree $2$
Conductor $379050$
Sign $-1$
Analytic cond. $3026.72$
Root an. cond. $55.0157$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 3·11-s − 12-s + 4·13-s + 14-s + 16-s − 18-s + 21-s + 3·22-s − 9·23-s + 24-s − 4·26-s − 27-s − 28-s + 3·29-s + 2·31-s − 32-s + 3·33-s + 36-s + 10·37-s − 4·39-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 − 0.904·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s + 0.639·22-s − 1.87·23-s + 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + 0.359·31-s − 0.176·32-s + 0.522·33-s + 1/6·36-s + 1.64·37-s − 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(379050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(3026.72\)
Root analytic conductor: \(55.0157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
19 \( 1 \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.81502649820538, −11.99257100011101, −11.76704423518807, −11.37227796139000, −10.80807158993283, −10.36458474398657, −10.03239958908395, −9.761068698884786, −9.045389067931086, −8.536805306318145, −8.235794355989974, −7.605660952066497, −7.426161567624092, −6.533607681095132, −6.263507083580006, −5.943438741201587, −5.404781230329834, −4.747693541488167, −4.211007677004945, −3.708655844607091, −3.011419848725195, −2.585337317995472, −1.835832352564830, −1.344630146404482, −0.5911756636439939, 0, 0.5911756636439939, 1.344630146404482, 1.835832352564830, 2.585337317995472, 3.011419848725195, 3.708655844607091, 4.211007677004945, 4.747693541488167, 5.404781230329834, 5.943438741201587, 6.263507083580006, 6.533607681095132, 7.426161567624092, 7.605660952066497, 8.235794355989974, 8.536805306318145, 9.045389067931086, 9.761068698884786, 10.03239958908395, 10.36458474398657, 10.80807158993283, 11.37227796139000, 11.76704423518807, 11.99257100011101, 12.81502649820538

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