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Properties

Label 1-6025-6025.1073-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.414 + 0.910i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (−0.0523 + 0.998i)2-s + (0.998 − 0.0523i)3-s + (−0.994 − 0.104i)4-s + i·6-s + (−0.333 − 0.942i)7-s + (0.156 − 0.987i)8-s + (0.994 − 0.104i)9-s + (−0.942 + 0.333i)11-s + (−0.998 − 0.0523i)12-s + (0.824 − 0.566i)13-s + (0.958 − 0.284i)14-s + (0.978 + 0.207i)16-s + (0.649 + 0.760i)17-s + (0.0523 + 0.998i)18-s + (−0.284 + 0.958i)19-s + ⋯
L(s)  = 1  + (−0.0523 + 0.998i)2-s + (0.998 − 0.0523i)3-s + (−0.994 − 0.104i)4-s + i·6-s + (−0.333 − 0.942i)7-s + (0.156 − 0.987i)8-s + (0.994 − 0.104i)9-s + (−0.942 + 0.333i)11-s + (−0.998 − 0.0523i)12-s + (0.824 − 0.566i)13-s + (0.958 − 0.284i)14-s + (0.978 + 0.207i)16-s + (0.649 + 0.760i)17-s + (0.0523 + 0.998i)18-s + (−0.284 + 0.958i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.414 + 0.910i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.414 + 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.988820124 + 1.280176528i\)
\(L(\frac12)\) \(\approx\) \(1.988820124 + 1.280176528i\)
\(L(1)\) \(\approx\) \(1.251378553 + 0.5150713294i\)
\(L(1)\) \(\approx\) \(1.251378553 + 0.5150713294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.0523 + 0.998i)T \)
3 \( 1 + (0.998 - 0.0523i)T \)
7 \( 1 + (-0.333 - 0.942i)T \)
11 \( 1 + (-0.942 + 0.333i)T \)
13 \( 1 + (0.824 - 0.566i)T \)
17 \( 1 + (0.649 + 0.760i)T \)
19 \( 1 + (-0.284 + 0.958i)T \)
23 \( 1 + (0.972 - 0.233i)T \)
29 \( 1 + (0.0523 - 0.998i)T \)
31 \( 1 + (0.182 + 0.983i)T \)
37 \( 1 + (0.999 - 0.0261i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.760 + 0.649i)T \)
47 \( 1 + (0.453 + 0.891i)T \)
53 \( 1 + (-0.838 - 0.544i)T \)
59 \( 1 + (-0.358 + 0.933i)T \)
61 \( 1 + (-0.987 - 0.156i)T \)
67 \( 1 + (0.544 + 0.838i)T \)
71 \( 1 + (-0.430 + 0.902i)T \)
73 \( 1 + (0.0784 - 0.996i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (-0.430 + 0.902i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−18.1308338934389825443543149934, −16.91631654454863245852359588289, −16.225955837031608524199499920354, −15.42926122266600970981595321575, −15.00448769269158489448792053549, −14.03280937262154697066688722893, −13.613663154115831349692787720806, −12.89783407517816733324812634189, −12.566731424178725135532353266351, −11.495702975004280241438271573455, −11.08297368966523248732735871949, −10.23282953493910810684447236764, −9.40432398345115659460354857624, −9.150870003823701716602817931765, −8.43989752098547788011066606478, −7.850686965990267112299929476647, −6.957103578205115016502542047659, −5.93313932748277846736354011233, −5.08041369139536589872976946561, −4.51942683963541745798122771894, −3.46522095921143878112337989294, −2.9923406552323125309532387048, −2.4669749647209623094754976157, −1.67430315904607833803210193097, −0.69507010966763921812912906034, 0.83596932963692583223442563796, 1.54354917092544835644251143044, 2.85869736840577653789723933818, 3.47509136886815265356818296526, 4.17170423953184031282392586552, 4.79731041947983003447151633463, 5.846563166884507932452069708989, 6.40322620626194733763378948747, 7.29623565583519890083448676639, 7.84202143513104656612449521822, 8.185873585164237732999086568635, 8.997135192627145254667107540245, 9.80458503625268084030196735469, 10.329013146822124979001682651675, 10.80200110007500441437731661693, 12.36336682143869423712150743293, 12.97484831903899378394374066800, 13.25445625972793063662866002273, 14.041160680580645888705627509427, 14.59555048481325520078049559985, 15.19028760089717489530206754516, 15.85555226904594863828723256139, 16.3397898343181604588161609745, 17.10448998238933298589072138133, 17.78082839235745530159333677848

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