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Properties

Label 1-35e2-1225.338-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.577 + 0.816i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (0.817 − 0.575i)2-s + (0.538 − 0.842i)3-s + (0.337 − 0.941i)4-s + (−0.0448 − 0.998i)6-s + (−0.266 − 0.963i)8-s + (−0.420 − 0.907i)9-s + (0.420 − 0.907i)11-s + (−0.611 − 0.791i)12-s + (−0.512 + 0.858i)13-s + (−0.772 − 0.635i)16-s + (0.762 + 0.646i)17-s + (−0.866 − 0.5i)18-s + (0.104 − 0.994i)19-s + (−0.178 − 0.983i)22-s + (−0.379 − 0.925i)23-s + (−0.955 − 0.294i)24-s + ⋯
L(s)  = 1  + (0.817 − 0.575i)2-s + (0.538 − 0.842i)3-s + (0.337 − 0.941i)4-s + (−0.0448 − 0.998i)6-s + (−0.266 − 0.963i)8-s + (−0.420 − 0.907i)9-s + (0.420 − 0.907i)11-s + (−0.611 − 0.791i)12-s + (−0.512 + 0.858i)13-s + (−0.772 − 0.635i)16-s + (0.762 + 0.646i)17-s + (−0.866 − 0.5i)18-s + (0.104 − 0.994i)19-s + (−0.178 − 0.983i)22-s + (−0.379 − 0.925i)23-s + (−0.955 − 0.294i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (338, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.309236018 - 2.530623938i\)
\(L(\frac12)\) \(\approx\) \(-1.309236018 - 2.530623938i\)
\(L(1)\) \(\approx\) \(1.108028691 - 1.374148630i\)
\(L(1)\) \(\approx\) \(1.108028691 - 1.374148630i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.817 - 0.575i)T \)
3 \( 1 + (0.538 - 0.842i)T \)
11 \( 1 + (0.420 - 0.907i)T \)
13 \( 1 + (-0.512 + 0.858i)T \)
17 \( 1 + (0.762 + 0.646i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (-0.379 - 0.925i)T \)
29 \( 1 + (-0.473 + 0.880i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.611 - 0.791i)T \)
41 \( 1 + (-0.550 - 0.834i)T \)
43 \( 1 + (-0.781 + 0.623i)T \)
47 \( 1 + (0.800 + 0.599i)T \)
53 \( 1 + (-0.941 - 0.337i)T \)
59 \( 1 + (0.163 - 0.986i)T \)
61 \( 1 + (-0.280 + 0.959i)T \)
67 \( 1 + (-0.406 - 0.913i)T \)
71 \( 1 + (0.983 - 0.178i)T \)
73 \( 1 + (-0.999 + 0.0149i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (-0.919 - 0.393i)T \)
89 \( 1 + (0.873 + 0.486i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

−21.53448408925872485637235198653, −20.63543230614412545945228503328, −20.34078641941910344488125224989, −19.413619417640584793269729355904, −18.252523126586018354135611579, −17.12988742684990959275194817594, −16.85204895107717358587381991924, −15.62256593358310380099726121546, −15.39709838321961570753881983989, −14.500914388913405055568053711422, −13.96711456135950598267033913713, −13.10478746802064307648207987026, −12.10182523282973971727427402456, −11.576913457793789332075089400328, −10.164919214548045747832560998645, −9.80287327773734156709810665134, −8.60152842312724745758647408176, −7.816660261430749543757673585462, −7.21063876635556525383157548923, −5.93250951748187914061214229808, −5.19004329282898084309845312617, −4.45665792167275089296992064111, −3.54837880180335198794718371521, −2.88076810556193271198492350691, −1.75934585316861622978777489811, 0.34618658837902909909373218724, 1.35281915982895819381798819884, 2.21900590891213572723221706055, 3.11792374742833021672661944003, 3.8738347731951072087440917374, 4.94075624735684519431151555062, 6.04542312906112687549988732368, 6.61890886841795311594741976697, 7.50845279197470517504769258579, 8.66500628210144468456611000169, 9.31097884997156570988920869850, 10.37304496326371477824445881207, 11.33553244694502366195466019868, 11.99792630652202265105072352682, 12.66872731108576116521521031792, 13.46212947398792904933783931457, 14.236888642941765510741807539878, 14.54101159308919796918595400022, 15.57646419166893573146140613476, 16.56192811948445270470541365984, 17.46782886607427797593736416659, 18.62818804225491623005652674784, 19.040196113328014946444432217085, 19.66257266135939210231220977724, 20.40013470159068036650446218584

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