↑

Properties

Label 2-1925-385.87-c0-0-3
Degree $2$
Conductor $1925$
Sign $0.635 - 0.772i$
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  + (1.67 + 0.448i)2-s + (1.73 + 1.00i)4-s + (0.707 + 0.707i)7-s + (1.22 + 1.22i)8-s + (−0.866 + 0.5i)9-s + (0.5 − 0.866i)11-s + (0.707 − 0.707i)13-s + (0.866 + 1.5i)14-s + (0.500 + 0.866i)16-s + (−1.93 + 0.517i)17-s + (−1.67 + 0.448i)18-s + (1.22 − 1.22i)22-s + (1.5 − 0.866i)26-s + (0.517 + 1.93i)28-s + (−1.5 − 0.866i)31-s + ⋯
L(s)  = 1  + (1.67 + 0.448i)2-s + (1.73 + 1.00i)4-s + (0.707 + 0.707i)7-s + (1.22 + 1.22i)8-s + (−0.866 + 0.5i)9-s + (0.5 − 0.866i)11-s + (0.707 − 0.707i)13-s + (0.866 + 1.5i)14-s + (0.500 + 0.866i)16-s + (−1.93 + 0.517i)17-s + (−1.67 + 0.448i)18-s + (1.22 − 1.22i)22-s + (1.5 − 0.866i)26-s + (0.517 + 1.93i)28-s + (−1.5 − 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.635 - 0.772i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.635 - 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.971975326\)
\(L(\frac12)\) \(\approx\) \(2.971975326\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
3 \( 1 + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 + (1.93 - 0.517i)T + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - iT^{2} \)
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

−9.183908707602577038118868620035, −8.503630509649949472000476129083, −7.88544092326504074486549241350, −6.78177233601868154119035476580, −5.86504675836584090930206336490, −5.69754289233737980978262323126, −4.66701824930478295777472328860, −3.88603043264046080313429175970, −2.90429332001443441581638697961, −2.05241525202785152831015242904, 1.63617700266161404225318005579, 2.51014234736039832204302109337, 3.77089606954649612671558263517, 4.22712424326536703263083496644, 5.00332464260445909467829663055, 5.89764800031116454216799108167, 6.79274121238127647448045190845, 7.19678908652664162937588123942, 8.695524394123927092815918709485, 9.175327442855281301291245746976

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