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Properties

Label 1-35e2-1225.106-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.340 - 0.940i$
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  + (−0.393 − 0.919i)2-s + (−0.550 − 0.834i)3-s + (−0.691 + 0.722i)4-s + (−0.550 + 0.834i)6-s + (0.936 + 0.351i)8-s + (−0.393 + 0.919i)9-s + (−0.393 − 0.919i)11-s + (0.983 + 0.178i)12-s + (0.753 − 0.657i)13-s + (−0.0448 − 0.998i)16-s + (−0.691 − 0.722i)17-s + 18-s + (0.309 + 0.951i)19-s + (−0.691 + 0.722i)22-s + (0.983 − 0.178i)23-s + (−0.222 − 0.974i)24-s + ⋯
L(s)  = 1  + (−0.393 − 0.919i)2-s + (−0.550 − 0.834i)3-s + (−0.691 + 0.722i)4-s + (−0.550 + 0.834i)6-s + (0.936 + 0.351i)8-s + (−0.393 + 0.919i)9-s + (−0.393 − 0.919i)11-s + (0.983 + 0.178i)12-s + (0.753 − 0.657i)13-s + (−0.0448 − 0.998i)16-s + (−0.691 − 0.722i)17-s + 18-s + (0.309 + 0.951i)19-s + (−0.691 + 0.722i)22-s + (0.983 − 0.178i)23-s + (−0.222 − 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.340 - 0.940i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.340 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5144822031 - 0.7335326634i\)
\(L(\frac12)\) \(\approx\) \(0.5144822031 - 0.7335326634i\)
\(L(1)\) \(\approx\) \(0.5700811956 - 0.4439548603i\)
\(L(1)\) \(\approx\) \(0.5700811956 - 0.4439548603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.393 - 0.919i)T \)
3 \( 1 + (-0.550 - 0.834i)T \)
11 \( 1 + (-0.393 - 0.919i)T \)
13 \( 1 + (0.753 - 0.657i)T \)
17 \( 1 + (-0.691 - 0.722i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.983 - 0.178i)T \)
29 \( 1 + (0.134 + 0.990i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.983 + 0.178i)T \)
41 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (0.858 + 0.512i)T \)
53 \( 1 + (-0.691 + 0.722i)T \)
59 \( 1 + (-0.0448 - 0.998i)T \)
61 \( 1 + (0.473 - 0.880i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (0.753 + 0.657i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.858 - 0.512i)T \)
89 \( 1 + (0.753 + 0.657i)T \)
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

−21.45648884289115948462098726537, −20.66365628162829808074038763861, −19.79719322702502124776692935680, −18.8511129371020800478445763129, −18.01918571266417157040831303112, −17.36016986370976645593332944345, −16.83431054960541038116023105973, −15.904832047036234722416725904593, −15.287461013641749331687185821473, −14.9350316020555643955709786784, −13.692755524863355830943034522593, −13.06777967095918553413204054914, −11.81319154588090545733751825306, −10.952932974269002963696055529098, −10.29576097324909680546560568950, −9.34851780590168469497060084329, −8.930091549568935465068647328080, −7.8263849837770889511131785153, −6.822614276480271303522358563897, −6.2084965365990179860961021268, −5.248761038159017002597243733466, −4.51182514507589683296993649246, −3.83178616813733494494780986270, −2.224402386956936069147936894705, −0.77515075612658770472461835252, 0.72386420998387903452032275807, 1.45292851164584956004405793366, 2.71550926499025772134074294422, 3.317708961708107052958971524417, 4.708041651904672712423786865539, 5.50686686385217258729038008569, 6.50457524976121188762830148245, 7.531476978605400187060322941904, 8.28075595806321807420111523549, 8.931436757931007025840002896791, 10.135330539278903693781823474626, 11.00730598139755474916373536277, 11.273304175438732685937184694143, 12.356623279450594526879906561046, 12.93470392779349207291474599355, 13.62768591606222235774855594316, 14.3081914306106009031823464449, 15.88477362079006964211491161375, 16.44728773863952074347026157980, 17.36339941904865767326143227492, 18.05781070500255802059330411474, 18.63386580195960153723390313610, 19.14413312656945710601720170422, 20.17455867007731883697757653108, 20.667484738955386976126755290439

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