↑

Properties

Label 1-35e2-1225.302-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.959 + 0.280i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (−0.512 + 0.858i)2-s + (0.266 + 0.963i)3-s + (−0.473 − 0.880i)4-s + (−0.963 − 0.266i)6-s + (0.998 + 0.0448i)8-s + (−0.858 + 0.512i)9-s + (0.858 + 0.512i)11-s + (0.722 − 0.691i)12-s + (0.0896 + 0.995i)13-s + (−0.550 + 0.834i)16-s + (0.880 + 0.473i)17-s − i·18-s + (0.809 − 0.587i)19-s + (−0.880 + 0.473i)22-s + (0.722 + 0.691i)23-s + (0.222 + 0.974i)24-s + ⋯
L(s)  = 1  + (−0.512 + 0.858i)2-s + (0.266 + 0.963i)3-s + (−0.473 − 0.880i)4-s + (−0.963 − 0.266i)6-s + (0.998 + 0.0448i)8-s + (−0.858 + 0.512i)9-s + (0.858 + 0.512i)11-s + (0.722 − 0.691i)12-s + (0.0896 + 0.995i)13-s + (−0.550 + 0.834i)16-s + (0.880 + 0.473i)17-s − i·18-s + (0.809 − 0.587i)19-s + (−0.880 + 0.473i)22-s + (0.722 + 0.691i)23-s + (0.222 + 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.959 + 0.280i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (302, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.959 + 0.280i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2888378674 + 2.015582873i\)
\(L(\frac12)\) \(\approx\) \(0.2888378674 + 2.015582873i\)
\(L(1)\) \(\approx\) \(0.6984781653 + 0.7448334784i\)
\(L(1)\) \(\approx\) \(0.6984781653 + 0.7448334784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.512 + 0.858i)T \)
3 \( 1 + (0.266 + 0.963i)T \)
11 \( 1 + (0.858 + 0.512i)T \)
13 \( 1 + (0.0896 + 0.995i)T \)
17 \( 1 + (0.880 + 0.473i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (0.722 + 0.691i)T \)
29 \( 1 + (-0.983 - 0.178i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.722 - 0.691i)T \)
41 \( 1 + (0.936 + 0.351i)T \)
43 \( 1 + (0.781 - 0.623i)T \)
47 \( 1 + (0.657 + 0.753i)T \)
53 \( 1 + (0.880 - 0.473i)T \)
59 \( 1 + (0.550 - 0.834i)T \)
61 \( 1 + (0.134 + 0.990i)T \)
67 \( 1 + (-0.587 - 0.809i)T \)
71 \( 1 + (0.473 + 0.880i)T \)
73 \( 1 + (-0.0896 + 0.995i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.657 - 0.753i)T \)
89 \( 1 + (0.995 + 0.0896i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.44924545220029745537436562768, −19.801040287488256835991719669132, −18.99883457545360839099129605815, −18.455956610898681223772581831591, −17.85306442438759673421149646547, −16.87216636741873162883903188237, −16.44126767533729458436174193616, −14.92165061451097108422880284903, −14.19703341834535623016440236319, −13.38325237859786446897253642934, −12.70444369669627745462447493394, −11.98515961233327172074870352854, −11.33661003421298133411776775609, −10.45028516916810324027288517579, −9.370727370167638307446713063374, −8.85069678385411638765778703804, −7.760768123761265218212740962445, −7.46014382134235736269243605747, −6.17856699212120997080050982281, −5.28121546845830074846034185170, −3.791650538549366548302833524614, −3.161037294811431546281080845536, −2.26693038185232942256557837840, −1.0874908537646193352662610547, −0.65374147411591756669192365824, 0.967811228204137310190130574589, 2.10588299189211176837436308331, 3.59807123233685288248431297081, 4.286634991274818396628553932614, 5.24397140111015922766884406716, 5.95068820471732381335459850381, 7.09577053761327456213243125849, 7.70042605470402062385802176275, 8.99048415246907516227223631361, 9.21381077157870025219859386445, 9.956606779935113786214617539303, 10.96208107481192953567688394693, 11.61850529027349798219259767321, 12.946378430475449624052963976751, 14.063367898290488001174796095595, 14.46506861294451446161472724886, 15.17789359398925258217828651290, 15.98765087848592828845737410920, 16.64469682680317019329086004060, 17.199444877117565519528184749168, 18.03613179779613657683297241428, 19.12570500397838243651439498412, 19.59044848209811406966361306232, 20.43196639493051420487169153497, 21.350973785152037727913816712069

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