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Properties

Label 1-245-245.32-r1-0-0
Degree $1$
Conductor $245$
Sign $-0.746 + 0.665i$
Analytic cond. $26.3289$
Root an. cond. $26.3289$
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  + (−0.149 − 0.988i)2-s + (0.563 − 0.826i)3-s + (−0.955 + 0.294i)4-s + (−0.900 − 0.433i)6-s + (0.433 + 0.900i)8-s + (−0.365 − 0.930i)9-s + (0.365 − 0.930i)11-s + (−0.294 + 0.955i)12-s + (0.781 − 0.623i)13-s + (0.826 − 0.563i)16-s + (−0.680 − 0.733i)17-s + (−0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + (−0.974 − 0.222i)22-s + (−0.680 + 0.733i)23-s + (0.988 + 0.149i)24-s + ⋯
L(s)  = 1  + (−0.149 − 0.988i)2-s + (0.563 − 0.826i)3-s + (−0.955 + 0.294i)4-s + (−0.900 − 0.433i)6-s + (0.433 + 0.900i)8-s + (−0.365 − 0.930i)9-s + (0.365 − 0.930i)11-s + (−0.294 + 0.955i)12-s + (0.781 − 0.623i)13-s + (0.826 − 0.563i)16-s + (−0.680 − 0.733i)17-s + (−0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + (−0.974 − 0.222i)22-s + (−0.680 + 0.733i)23-s + (0.988 + 0.149i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(26.3289\)
Root analytic conductor: \(26.3289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (1:\ ),\ -0.746 + 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4922018155 - 1.290573920i\)
\(L(\frac12)\) \(\approx\) \(-0.4922018155 - 1.290573920i\)
\(L(1)\) \(\approx\) \(0.5866918602 - 0.8385409236i\)
\(L(1)\) \(\approx\) \(0.5866918602 - 0.8385409236i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.149 - 0.988i)T \)
3 \( 1 + (0.563 - 0.826i)T \)
11 \( 1 + (0.365 - 0.930i)T \)
13 \( 1 + (0.781 - 0.623i)T \)
17 \( 1 + (-0.680 - 0.733i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.680 + 0.733i)T \)
29 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.294 + 0.955i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (-0.149 - 0.988i)T \)
53 \( 1 + (-0.294 - 0.955i)T \)
59 \( 1 + (-0.0747 - 0.997i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.149 + 0.988i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.781 - 0.623i)T \)
89 \( 1 + (-0.365 - 0.930i)T \)
97 \( 1 + iT \)
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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

−26.40782863726576710970693920293, −25.55834409113506836277280459852, −24.911907000950310969250277023212, −23.783899108288683879742417319537, −22.759746149381456507814132922230, −22.04971295561755594804173965198, −20.95167686235569951394090708165, −19.955567895026869036560352896235, −18.96083574497419203695070930193, −17.88181514776596556951651685343, −16.86850169975294951650138022273, −16.01945443636208589098050431652, −15.263764170478455888221539443686, −14.35728277745828719596722747059, −13.66692098187592645092742202083, −12.340605744216703352679838188887, −10.717903190293839913355130330441, −9.81257169576704182488396395343, −8.89735915003160919754114700137, −8.096719570858516593090059258125, −6.87073714094045248417868481872, −5.72085489201821393463039061966, −4.43607221248415026332013970372, −3.77274945346100614469991916135, −1.82105287550739040217873096383, 0.442708475348373610906048795048, 1.54682270386224917434185559557, 2.88775343459698126136582512240, 3.66216974391898119620262005321, 5.31205438317921052315147849060, 6.703325391361660255629625424411, 8.05354902092606541843313275982, 8.77481669030515589495245889500, 9.7444301774173767866898369440, 11.21390223142901961113787883643, 11.75149668441295101164887755043, 13.16555494586468465488746214323, 13.48869897615050204105595884439, 14.52353719828670283087373664929, 15.91826114279833268126040426840, 17.34751811686933756646824952723, 18.19438329784556281017872588826, 18.807372980141351539065258143787, 19.99298242064859675335608458587, 20.24265653178613034702067752540, 21.539794003921368307404833339137, 22.38825948925138322508501331479, 23.49855897801482024485206553728, 24.31820833238345080954775889205, 25.45532512988789338130466576971

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