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Properties

Label 1-245-245.173-r0-0-0
Degree $1$
Conductor $245$
Sign $0.665 - 0.746i$
Analytic cond. $1.13777$
Root an. cond. $1.13777$
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

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

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.665 - 0.746i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ 0.665 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5836713369 - 0.2613828001i\)
\(L(\frac12)\) \(\approx\) \(0.5836713369 - 0.2613828001i\)
\(L(1)\) \(\approx\) \(0.6937819262 + 0.02369565930i\)
\(L(1)\) \(\approx\) \(0.6937819262 + 0.02369565930i\)

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

−26.57931325640276600456239023175, −25.71060680396717048276370016252, −24.067406585032764226248002041570, −23.24218891893175896365738110369, −22.20881301035547320737115293045, −21.55350691029343419489284841905, −20.97929496408588889696937566684, −19.725404011736750956559268379955, −19.024733004975450507776034485625, −17.82042754679069467115835619313, −16.935965536312881855596307843602, −16.24474862947417043044641103384, −14.70919600962241732573727717304, −13.99502582737099930870863971001, −12.47956116537590542686316945545, −11.82887307078318339801088926881, −10.83090812475085001824834287369, −10.04260641307371802604499235505, −9.13502005180339595614831494986, −8.083753949163400149385424851476, −6.25836920494706664040638865618, −5.131180860306525836971717455638, −4.02970238811959793425773192360, −3.148813524545414912106286818621, −1.40931431350673020839234839096, 0.55853271510574884554848987329, 2.321776370505951722943343444382, 4.39836779932255772792524947496, 5.35706439232515096760302906460, 6.433194289842565583828963329688, 7.314123177310271741755269173801, 8.05538921986595960806054639718, 9.4354461523533538954613776701, 10.43437292396856300750690694000, 11.94291511199967408240330929555, 12.73217085522451840555969817425, 13.75356755367860703781663747770, 14.711277521616044505830078300698, 15.71437949514602233747704215219, 16.89784222712536273355299505204, 17.45343270639935327217641089678, 18.27469645548373153638995202915, 19.20486237962078746794297506459, 20.15333723784357484076450465908, 21.83346618063462485293721850410, 22.74533948357421152270370022869, 23.213687393861298330000245014124, 24.37914864208981361630609311232, 24.88575616107281381486516415217, 25.732608393146894218504385457387

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