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Properties

Label 1-245-245.44-r0-0-0
Degree $1$
Conductor $245$
Sign $-0.961 - 0.274i$
Analytic cond. $1.13777$
Root an. cond. $1.13777$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.955 + 0.294i)2-s + (−0.365 − 0.930i)3-s + (0.826 − 0.563i)4-s + (0.623 + 0.781i)6-s + (−0.623 + 0.781i)8-s + (−0.733 + 0.680i)9-s + (−0.733 − 0.680i)11-s + (−0.826 − 0.563i)12-s + (0.222 − 0.974i)13-s + (0.365 − 0.930i)16-s + (−0.0747 + 0.997i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.900 + 0.433i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)24-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)2-s + (−0.365 − 0.930i)3-s + (0.826 − 0.563i)4-s + (0.623 + 0.781i)6-s + (−0.623 + 0.781i)8-s + (−0.733 + 0.680i)9-s + (−0.733 − 0.680i)11-s + (−0.826 − 0.563i)12-s + (0.222 − 0.974i)13-s + (0.365 − 0.930i)16-s + (−0.0747 + 0.997i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.900 + 0.433i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.961 - 0.274i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ -0.961 - 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04120539493 - 0.2947148365i\)
\(L(\frac12)\) \(\approx\) \(0.04120539493 - 0.2947148365i\)
\(L(1)\) \(\approx\) \(0.4509783621 - 0.1766386103i\)
\(L(1)\) \(\approx\) \(0.4509783621 - 0.1766386103i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.955 + 0.294i)T \)
3 \( 1 + (-0.365 - 0.930i)T \)
11 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 + (-0.0747 + 0.997i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (0.826 + 0.563i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.4981465871426001750396240925, −26.03275315717529224037156399681, −25.031426024982231328037515853140, −23.71649140570556867554465185465, −22.74875391005660981517220418210, −21.61178142139715143316533938407, −20.87292183824158497298437186071, −20.26142592678215767574943184684, −18.98712040607903843570528380569, −18.11785688068635717082730334982, −17.16370520465532393346075457044, −16.33460181524330895655500406829, −15.61035075331175666063183769957, −14.59130701043032319588118796842, −13.015133664774365674106654891084, −11.72548770044218671547980829237, −11.176000854880029770531315747758, −9.92023966711824023295781988307, −9.500090203488096617456655582444, −8.26584076332642012251796781220, −7.10065181236827571828684708216, −5.85513017622602409971426438264, −4.478192770924112431374503901751, −3.27624182751416664899021334148, −1.86451527748721283088292123108, 0.28451942339195964877741077706, 1.761561250269444397340561196491, 2.986828822152783839443703809362, 5.28940993104309710154418966290, 6.129567660705705641821398859312, 7.15848842660984089675718704361, 8.151625254214559927325586857886, 8.83101554809838508061494794297, 10.63242150181542065871905750919, 10.87169422886506694669037901310, 12.33642348770803344916220740532, 13.17894769541828207953756138174, 14.464349343668213641315894137957, 15.57317066214783571969443034585, 16.56086395460507967444096895891, 17.47620708595470616934626393296, 18.16932726096772646396750497764, 19.01619120387813810049696070162, 19.77073263480578825440217348923, 20.79671770274687945449729612568, 22.10913079069180507292421459010, 23.36184768768594886096952326353, 24.0217919468812920779443479778, 24.767659136845824914007562301252, 25.7516781028045952347093909967

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