↑

Properties

Label 2-1925-77.20-c0-0-5
Degree $2$
Conductor $1925$
Sign $0.998 + 0.0589i$
Analytic cond. $0.960700$
Root an. cond. $0.980153$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0646 − 0.198i)2-s + (0.773 + 0.562i)4-s + (0.809 + 0.587i)7-s + (0.330 − 0.240i)8-s + (0.309 − 0.951i)9-s + (0.669 − 0.743i)11-s + (0.169 − 0.122i)14-s + (0.269 + 0.828i)16-s + (−0.169 − 0.122i)18-s + (−0.104 − 0.181i)22-s − 1.82·23-s + (0.295 + 0.909i)28-s + (−1.08 − 0.786i)29-s + 0.591·32-s + (0.773 − 0.562i)36-s + (−0.169 − 0.122i)37-s + ⋯
L(s)  = 1  + (0.0646 − 0.198i)2-s + (0.773 + 0.562i)4-s + (0.809 + 0.587i)7-s + (0.330 − 0.240i)8-s + (0.309 − 0.951i)9-s + (0.669 − 0.743i)11-s + (0.169 − 0.122i)14-s + (0.269 + 0.828i)16-s + (−0.169 − 0.122i)18-s + (−0.104 − 0.181i)22-s − 1.82·23-s + (0.295 + 0.909i)28-s + (−1.08 − 0.786i)29-s + 0.591·32-s + (0.773 − 0.562i)36-s + (−0.169 − 0.122i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.998 + 0.0589i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.998 + 0.0589i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.623013209\)
\(L(\frac12)\) \(\approx\) \(1.623013209\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
good2 \( 1 + (-0.0646 + 0.198i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + 1.82T + T^{2} \)
29 \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.169 + 0.122i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 1.33T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 - 1.95T + T^{2} \)
71 \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.604 - 1.86i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.382639470732067947904105700279, −8.425896363333458882057320171390, −7.980916233454109023333844894565, −6.96738084142045847892467928814, −6.24277951998147675912961938144, −5.54815344876001850555602288586, −4.13219124570153606557266028032, −3.62794216580748110360600334705, −2.43238771484400248188681301348, −1.47816987759814060082487158590, 1.62460081381029028952870349235, 2.06878594978628535704041965203, 3.69673803620811370277539787954, 4.67337325199179855793790933412, 5.30338822524202034682682786062, 6.31833833565507466033160051518, 7.12062300431098646291229759466, 7.67750040125335758208283979931, 8.377608879511019067208910407267, 9.637545795341655803259449879515

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