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Properties

Label 1-35e2-1225.142-r1-0-0
Degree $1$
Conductor $1225$
Sign $0.175 - 0.984i$
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  + (−0.800 − 0.599i)2-s + (−0.0598 − 0.998i)3-s + (0.280 + 0.959i)4-s + (−0.550 + 0.834i)6-s + (0.351 − 0.936i)8-s + (−0.992 + 0.119i)9-s + (0.992 + 0.119i)11-s + (0.941 − 0.337i)12-s + (−0.657 − 0.753i)13-s + (−0.842 + 0.538i)16-s + (0.237 + 0.971i)17-s + (0.866 + 0.5i)18-s + (−0.669 + 0.743i)19-s + (−0.722 − 0.691i)22-s + (−0.762 + 0.646i)23-s + (−0.955 − 0.294i)24-s + ⋯
L(s)  = 1  + (−0.800 − 0.599i)2-s + (−0.0598 − 0.998i)3-s + (0.280 + 0.959i)4-s + (−0.550 + 0.834i)6-s + (0.351 − 0.936i)8-s + (−0.992 + 0.119i)9-s + (0.992 + 0.119i)11-s + (0.941 − 0.337i)12-s + (−0.657 − 0.753i)13-s + (−0.842 + 0.538i)16-s + (0.237 + 0.971i)17-s + (0.866 + 0.5i)18-s + (−0.669 + 0.743i)19-s + (−0.722 − 0.691i)22-s + (−0.762 + 0.646i)23-s + (−0.955 − 0.294i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ 0.175 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7537248381 - 0.6314444062i\)
\(L(\frac12)\) \(\approx\) \(0.7537248381 - 0.6314444062i\)
\(L(1)\) \(\approx\) \(0.5877739444 - 0.3220937013i\)
\(L(1)\) \(\approx\) \(0.5877739444 - 0.3220937013i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.800 - 0.599i)T \)
3 \( 1 + (-0.0598 - 0.998i)T \)
11 \( 1 + (0.992 + 0.119i)T \)
13 \( 1 + (-0.657 - 0.753i)T \)
17 \( 1 + (0.237 + 0.971i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (-0.762 + 0.646i)T \)
29 \( 1 + (-0.134 - 0.990i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (0.941 - 0.337i)T \)
41 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (0.781 - 0.623i)T \)
47 \( 1 + (0.999 + 0.0149i)T \)
53 \( 1 + (-0.959 + 0.280i)T \)
59 \( 1 + (-0.887 - 0.460i)T \)
61 \( 1 + (-0.999 + 0.0299i)T \)
67 \( 1 + (0.207 - 0.978i)T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (0.323 + 0.946i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.512 - 0.858i)T \)
89 \( 1 + (-0.193 + 0.981i)T \)
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

−21.03116795676077060101398160861, −20.06738163679634907996473099596, −19.741879212173615947996856829643, −18.72957102590870943320802102404, −17.91426356389068287731351593494, −16.977695702374151105137465949197, −16.615448508006528075996491100626, −15.92484459761725506891748576426, −15.00773603122734326077182715150, −14.41801676262706293079517085066, −13.8633098591593728252327514032, −12.26338613126996434189089450109, −11.43498449932508402723652587965, −10.78520228638362197941326516507, −9.871068605845981343933738482507, −9.16587602133062942147874459411, −8.7937995152246172355728695460, −7.58814978876414508999787467666, −6.71653675055216131636903022179, −5.95304845695898691831622385766, −4.89341183251253586167044367073, −4.30501922850229067233006408235, −2.9800678900547538855202392760, −1.87732288583592949291530832700, −0.50194588670505446005987558628, 0.49764695318710865327265862911, 1.583578248501334453168741218468, 2.17282555326819401088166966364, 3.33542832294871465787146756183, 4.17988283229278579697609799974, 5.77623186263790935544132961428, 6.435485113800788971289329720734, 7.56418935163070662890685653433, 7.91960474403273867096578048638, 8.877399741363303808017447928046, 9.713223771681162158892844534413, 10.61131253789216313840207739343, 11.42887976849494921649409518015, 12.30842927027995967966913743982, 12.58491150444391381034424774018, 13.57525472915021810170566522616, 14.51395992956410649685431514182, 15.37225097579225568670379137531, 16.6963310218390745271848765142, 17.15566640848107685113900200314, 17.68979674847048615458142939755, 18.65807198736850317367968008165, 19.15103361459214120478997105622, 19.95519148747721916183638270592, 20.30218043736389735152404489532

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