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Properties

Label 1-5e2-25.3-r1-0-0
Degree $1$
Conductor $25$
Sign $0.248 - 0.968i$
Analytic cond. $2.68662$
Root an. cond. $2.68662$
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  + (−0.587 + 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (0.309 − 0.951i)6-s − i·7-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.587 − 0.809i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + i·18-s + (−0.309 + 0.951i)19-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (0.309 − 0.951i)6-s − i·7-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.587 − 0.809i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + i·18-s + (−0.309 + 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.248 - 0.968i$
Analytic conductor: \(2.68662\)
Root analytic conductor: \(2.68662\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 25,\ (1:\ ),\ 0.248 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3058709774 - 0.2372578502i\)
\(L(\frac12)\) \(\approx\) \(0.3058709774 - 0.2372578502i\)
\(L(1)\) \(\approx\) \(0.4848512120 + 0.02760602250i\)
\(L(1)\) \(\approx\) \(0.4848512120 + 0.02760602250i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−38.560399837280829116862981053247, −37.06307838299446252015000881138, −35.80960056672587348378694907764, −34.78142326309883418651212081123, −33.7321194257003710551920296391, −31.547321921431999773449740222682, −30.4591037814164988461560766139, −28.88819627657530172206580130683, −28.47703090611749816448521120021, −27.09540874190338327240252041075, −25.52406529140324674648764539033, −23.85584942075733104719855047068, −22.220954773652521227645302647510, −21.372830439383272431502404615973, −19.46763958190183144695204980447, −18.282761273156686249270877059514, −17.30882065413628364207598943123, −15.70375978327536940787543772921, −13.05600052478992183658929999412, −11.97914172099989670851315075138, −10.740664138904519704033883926859, −9.07754959898039399954425069144, −7.11896130669450051008926366492, −4.90189089629268277047822372681, −2.14844988942813312364228150697, 0.39644555033453620608135353368, 4.70026480711542061894887985631, 6.26631616703937676868385893554, 7.81910086527958365643836925181, 9.95131118427465779841704816020, 10.96026688678815330242805263860, 13.263967648471832601029059351472, 15.13970570819968589009419891233, 16.46563996392816622255218531425, 17.37530766536392588808458507484, 18.68394949104805753415221212631, 20.497444812080845056448426103645, 22.479158317839920244834698576315, 23.49009302667954246447631091283, 24.640176484335156966507978799951, 26.5403620998003768679336352229, 27.16399880243125666591279078673, 28.654241666500588188044863021371, 29.651770418372795235961329998391, 31.96836609657264176240906051068, 33.14658241292956816995184174647, 34.02714081331372458594774403030, 35.124733557385188846998445571937, 36.29671920187941390681171293440, 37.60513145579877694390295059206

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