Properties

Label 2-257754-1.1-c1-0-16
Degree $2$
Conductor $257754$
Sign $1$
Analytic cond. $2058.17$
Root an. cond. $45.3671$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 7-s − 8-s + 9-s + 2·10-s + 12-s + 6·13-s + 14-s − 2·15-s + 16-s + 17-s − 18-s − 2·20-s − 21-s − 8·23-s − 24-s − 25-s − 6·26-s + 27-s − 28-s + 6·29-s + 2·30-s + 8·31-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 + 0.288·12-s + 1.66·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.447·20-s − 0.218·21-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 + 1.11·29-s + 0.365·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(257754\)    =    \(2 \cdot 3 \cdot 7 \cdot 17 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2058.17\)
Root analytic conductor: \(45.3671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{257754} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 257754,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.075858513\)
\(L(\frac12)\) \(\approx\) \(2.075858513\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−12.70794133258610, −12.21713771986754, −11.84357035005775, −11.57130724016927, −10.79645438138171, −10.37887647286531, −10.22822131607696, −9.435915432047634, −9.058129697283077, −8.513469303690342, −8.196784492167157, −7.872343195310616, −7.336799360140015, −6.728875360644758, −6.302983707934138, −5.849841444254471, −5.245188113070527, −4.241641906568288, −4.000697418406772, −3.631802005353173, −2.880209948354235, −2.463315036410687, −1.688210377917721, −1.013482199951452, −0.4972747967542780, 0.4972747967542780, 1.013482199951452, 1.688210377917721, 2.463315036410687, 2.880209948354235, 3.631802005353173, 4.000697418406772, 4.241641906568288, 5.245188113070527, 5.849841444254471, 6.302983707934138, 6.728875360644758, 7.336799360140015, 7.872343195310616, 8.196784492167157, 8.513469303690342, 9.058129697283077, 9.435915432047634, 10.22822131607696, 10.37887647286531, 10.79645438138171, 11.57130724016927, 11.84357035005775, 12.21713771986754, 12.70794133258610

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