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Properties

Label 1-35e2-1225.3-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.932 - 0.360i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (0.981 − 0.193i)2-s + (−0.986 + 0.163i)3-s + (0.925 − 0.379i)4-s + (−0.936 + 0.351i)6-s + (0.834 − 0.550i)8-s + (0.946 − 0.323i)9-s + (−0.946 − 0.323i)11-s + (−0.850 + 0.525i)12-s + (−0.919 + 0.393i)13-s + (0.712 − 0.701i)16-s + (0.611 + 0.791i)17-s + (0.866 − 0.5i)18-s + (0.669 + 0.743i)19-s + (−0.990 − 0.134i)22-s + (0.0299 − 0.999i)23-s + (−0.733 + 0.680i)24-s + ⋯
L(s)  = 1  + (0.981 − 0.193i)2-s + (−0.986 + 0.163i)3-s + (0.925 − 0.379i)4-s + (−0.936 + 0.351i)6-s + (0.834 − 0.550i)8-s + (0.946 − 0.323i)9-s + (−0.946 − 0.323i)11-s + (−0.850 + 0.525i)12-s + (−0.919 + 0.393i)13-s + (0.712 − 0.701i)16-s + (0.611 + 0.791i)17-s + (0.866 − 0.5i)18-s + (0.669 + 0.743i)19-s + (−0.990 − 0.134i)22-s + (0.0299 − 0.999i)23-s + (−0.733 + 0.680i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.932 - 0.360i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.932 - 0.360i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.050606672 - 0.3820976821i\)
\(L(\frac12)\) \(\approx\) \(2.050606672 - 0.3820976821i\)
\(L(1)\) \(\approx\) \(1.451379007 - 0.1606329411i\)
\(L(1)\) \(\approx\) \(1.451379007 - 0.1606329411i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.981 - 0.193i)T \)
3 \( 1 + (-0.986 + 0.163i)T \)
11 \( 1 + (-0.946 - 0.323i)T \)
13 \( 1 + (-0.919 + 0.393i)T \)
17 \( 1 + (0.611 + 0.791i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.0299 - 0.999i)T \)
29 \( 1 + (0.691 + 0.722i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.850 - 0.525i)T \)
41 \( 1 + (0.0448 - 0.998i)T \)
43 \( 1 + (-0.781 + 0.623i)T \)
47 \( 1 + (0.907 + 0.420i)T \)
53 \( 1 + (-0.379 - 0.925i)T \)
59 \( 1 + (0.251 + 0.967i)T \)
61 \( 1 + (0.646 - 0.762i)T \)
67 \( 1 + (0.207 + 0.978i)T \)
71 \( 1 + (0.134 - 0.990i)T \)
73 \( 1 + (-0.119 - 0.992i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.0896 + 0.995i)T \)
89 \( 1 + (-0.599 - 0.800i)T \)
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

−21.4982432123997745055799078351, −20.605015752474983749162393273632, −19.82675992914370746824783401620, −18.838939542654148905387868069204, −17.88491514699054504433802252325, −17.26824450540496906908312387164, −16.48461751492491995055420973690, −15.58138903214744515480991388227, −15.31398659331207983474645960031, −14.01329001441089721669505201383, −13.35863537468861861989338187488, −12.62688757172709488435194039834, −11.83516759917126778823581593416, −11.3965328638633104372542604936, −10.273881635250041177909935512974, −9.75201297115146771113813237332, −7.97825553902746307825914823097, −7.42341392573019668225605845404, −6.68329712615786727705245780877, −5.63899220717022576768095636427, −5.072063144130163129934282141244, −4.486493501367019055682091896890, −3.10658014607629170074030404839, −2.341879445405773204333529880291, −0.95061114862914829047514256092, 0.883244157225026411227898410451, 2.09445557786739889691942079907, 3.13976023680972671887291618952, 4.18297542057380363818445296025, 4.94898143968970683142683495406, 5.63201852383272658681195464250, 6.3822987447634331928157730954, 7.26224820145145200128636865334, 8.130495700913015896944938392981, 9.730325222740944091752537030671, 10.34057428413675183856476541221, 10.96178273451092253894581918010, 11.97185307801232227752528257937, 12.38869060988788767934041597593, 13.13081049493911507334162531855, 14.15728752957992581854987781969, 14.8431342189844495539306162008, 15.75315781744633011302874805944, 16.39188844766272632110755814671, 16.96707893880311438338852373178, 18.05983168648130917307456595741, 18.85703819365679760768153992561, 19.58442184420550790059451741805, 20.78075787700332384663176094644, 21.13717294044664989519501054070

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