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Properties

Label 1-35e2-1225.771-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.999 + 0.00512i$
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  + (0.983 + 0.178i)2-s + (−0.995 + 0.0896i)3-s + (0.936 + 0.351i)4-s + (−0.995 − 0.0896i)6-s + (0.858 + 0.512i)8-s + (0.983 − 0.178i)9-s + (0.983 + 0.178i)11-s + (−0.963 − 0.266i)12-s + (0.473 − 0.880i)13-s + (0.753 + 0.657i)16-s + (0.936 − 0.351i)17-s + 18-s + (0.309 − 0.951i)19-s + (0.936 + 0.351i)22-s + (−0.963 + 0.266i)23-s + (−0.900 − 0.433i)24-s + ⋯
L(s)  = 1  + (0.983 + 0.178i)2-s + (−0.995 + 0.0896i)3-s + (0.936 + 0.351i)4-s + (−0.995 − 0.0896i)6-s + (0.858 + 0.512i)8-s + (0.983 − 0.178i)9-s + (0.983 + 0.178i)11-s + (−0.963 − 0.266i)12-s + (0.473 − 0.880i)13-s + (0.753 + 0.657i)16-s + (0.936 − 0.351i)17-s + 18-s + (0.309 − 0.951i)19-s + (0.936 + 0.351i)22-s + (−0.963 + 0.266i)23-s + (−0.900 − 0.433i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.00512i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (771, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.999 + 0.00512i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.475968533 + 0.006349797230i\)
\(L(\frac12)\) \(\approx\) \(2.475968533 + 0.006349797230i\)
\(L(1)\) \(\approx\) \(1.644920961 + 0.1136115209i\)
\(L(1)\) \(\approx\) \(1.644920961 + 0.1136115209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.983 + 0.178i)T \)
3 \( 1 + (-0.995 + 0.0896i)T \)
11 \( 1 + (0.983 + 0.178i)T \)
13 \( 1 + (0.473 - 0.880i)T \)
17 \( 1 + (0.936 - 0.351i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.963 + 0.266i)T \)
29 \( 1 + (-0.550 + 0.834i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.963 - 0.266i)T \)
41 \( 1 + (-0.393 - 0.919i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.691 - 0.722i)T \)
53 \( 1 + (0.936 + 0.351i)T \)
59 \( 1 + (0.753 + 0.657i)T \)
61 \( 1 + (-0.0448 - 0.998i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.936 + 0.351i)T \)
73 \( 1 + (0.473 + 0.880i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.691 + 0.722i)T \)
89 \( 1 + (0.473 + 0.880i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−21.28842154519561784577511138944, −20.721948486702487580207004953594, −19.53876323957008652473275801331, −18.9870538940575603606438012801, −18.11101384458332595588032570822, −16.99967333758042549794948316675, −16.4408787374764516576651483217, −15.91505604892443268774198549389, −14.75336516428121076173577260391, −14.15524566714222172265866931017, −13.329493370729259734526194628957, −12.37328477678849122392326225525, −11.82336503798044653772449297858, −11.34414536310767351539695855611, −10.27538863833720048918843023432, −9.72434481873477161073257429306, −8.25647486567943883652764377343, −7.221473979579950968648909958362, −6.31985857915336354447572416755, −5.967337177653910620786434535518, −4.93393103773457053788228281421, −4.06559495829360574412066285238, −3.42907853433849822243758797690, −1.82401959420834188210446406807, −1.23853373430453691551311456988, 0.9422612554741524612049614924, 2.06677916712829734987290644243, 3.483792388979803405895331839425, 4.008509317625452051890118399885, 5.2724288260312592369834116251, 5.52766495593247518780802913411, 6.63118729129196999633685003421, 7.16365032819928084863849880414, 8.19062073893872594540118526824, 9.50747451217622684475810530691, 10.41093467913924825789441732662, 11.18440957241012313396618094436, 11.938573432337254635727841648283, 12.412413480021121295196470890120, 13.37469886293022097375365084298, 14.06374440252382233193436688146, 15.12433025855325353136195507604, 15.63908262043512501904800929039, 16.49155305700725224606385219872, 17.108547267273099786264995025561, 17.84601537435532328862824149930, 18.77311694739458581536776771666, 19.89660326893015393111838369335, 20.49113291719364831275531163338, 21.41423690583262101357075230947

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