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Properties

Label 1-6025-6025.166-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.496 + 0.868i$
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  + (0.207 − 0.978i)2-s + (−0.207 − 0.978i)3-s + (−0.913 − 0.406i)4-s − 6-s + (−0.838 + 0.544i)7-s + (−0.587 + 0.809i)8-s + (−0.913 + 0.406i)9-s + (−0.838 + 0.544i)11-s + (−0.207 + 0.978i)12-s + (0.998 − 0.0523i)13-s + (0.358 + 0.933i)14-s + (0.669 + 0.743i)16-s + (0.891 − 0.453i)17-s + (0.207 + 0.978i)18-s + (−0.358 + 0.933i)19-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (−0.207 − 0.978i)3-s + (−0.913 − 0.406i)4-s − 6-s + (−0.838 + 0.544i)7-s + (−0.587 + 0.809i)8-s + (−0.913 + 0.406i)9-s + (−0.838 + 0.544i)11-s + (−0.207 + 0.978i)12-s + (0.998 − 0.0523i)13-s + (0.358 + 0.933i)14-s + (0.669 + 0.743i)16-s + (0.891 − 0.453i)17-s + (0.207 + 0.978i)18-s + (−0.358 + 0.933i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.496 + 0.868i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.496 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03736152422 - 0.06440947527i\)
\(L(\frac12)\) \(\approx\) \(-0.03736152422 - 0.06440947527i\)
\(L(1)\) \(\approx\) \(0.5940937570 - 0.4087570967i\)
\(L(1)\) \(\approx\) \(0.5940937570 - 0.4087570967i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.207 - 0.978i)T \)
3 \( 1 + (-0.207 - 0.978i)T \)
7 \( 1 + (-0.838 + 0.544i)T \)
11 \( 1 + (-0.838 + 0.544i)T \)
13 \( 1 + (0.998 - 0.0523i)T \)
17 \( 1 + (0.891 - 0.453i)T \)
19 \( 1 + (-0.358 + 0.933i)T \)
23 \( 1 + (-0.987 + 0.156i)T \)
29 \( 1 + (-0.207 + 0.978i)T \)
31 \( 1 + (-0.0523 + 0.998i)T \)
37 \( 1 + (-0.777 - 0.629i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.891 + 0.453i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.743 + 0.669i)T \)
59 \( 1 + (0.994 - 0.104i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (-0.743 + 0.669i)T \)
71 \( 1 + (-0.544 - 0.838i)T \)
73 \( 1 + (-0.453 - 0.891i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (-0.544 - 0.838i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−17.8474863174501322366271905269, −17.251100194872697134085515836512, −16.57827108115789651692074461590, −16.23352557708403574812489540415, −15.47515685280099681254685058597, −15.27775422301968029525424520864, −14.24227712852823305343031577881, −13.58943493227868048240428286021, −13.2605936629103184926219368134, −12.32196868179338402111791183357, −11.53886639510850463005221216113, −10.59559453796766530263876032371, −10.16576789822367657059563651748, −9.49702073152759642706752141518, −8.682592393294904043215526494379, −8.211789615484222488611247133309, −7.37721684986314747961724995080, −6.46759529808603103320746987976, −5.91474950756249776715449926463, −5.44833179234882336605416945178, −4.5066967715388179236797409509, −3.78727536280675011068952089567, −3.45838745116137298768884937809, −2.48532182673719382461042912788, −0.75815354950586591886887338401, 0.025068885186912923494997270557, 1.26018481781575708111889698151, 1.80127680902494186379787065473, 2.71290904295295514736010906550, 3.22457743927500103436432184778, 4.02159347142939357274015885765, 5.22176950350670324342157179861, 5.62791455884014143734208259593, 6.26976667244214542029606739913, 7.17022346602916114490867965188, 8.02078597018595425395273574677, 8.61534764675257836300460722041, 9.3155784954091908066237057661, 10.31122215697181084430800713927, 10.50973308218806673739511464221, 11.5732585859277286861259127754, 12.2269145636165413278675802188, 12.45885001489726246078582941890, 13.23059564124061066849493831143, 13.6704290035115642900256341125, 14.43217754597508776201706058761, 15.12161636853370987375452980253, 16.16896783113017876140790532409, 16.53738512234812189257139703329, 17.72283694056686680671627524729

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