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Properties

Label 1-980-980.267-r0-0-0
Degree $1$
Conductor $980$
Sign $-0.999 - 0.0406i$
Analytic cond. $4.55110$
Root an. cond. $4.55110$
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  + (0.433 − 0.900i)3-s + (−0.623 − 0.781i)9-s + (−0.623 + 0.781i)11-s + (−0.781 − 0.623i)13-s + (−0.974 − 0.222i)17-s + 19-s + (0.974 − 0.222i)23-s + (−0.974 + 0.222i)27-s + (0.222 − 0.974i)29-s − 31-s + (0.433 + 0.900i)33-s + (−0.974 − 0.222i)37-s + (−0.900 + 0.433i)39-s + (−0.900 − 0.433i)41-s + (−0.433 − 0.900i)43-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)3-s + (−0.623 − 0.781i)9-s + (−0.623 + 0.781i)11-s + (−0.781 − 0.623i)13-s + (−0.974 − 0.222i)17-s + 19-s + (0.974 − 0.222i)23-s + (−0.974 + 0.222i)27-s + (0.222 − 0.974i)29-s − 31-s + (0.433 + 0.900i)33-s + (−0.974 − 0.222i)37-s + (−0.900 + 0.433i)39-s + (−0.900 − 0.433i)41-s + (−0.433 − 0.900i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.999 - 0.0406i$
Analytic conductor: \(4.55110\)
Root analytic conductor: \(4.55110\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 980,\ (0:\ ),\ -0.999 - 0.0406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01427318236 - 0.7019561792i\)
\(L(\frac12)\) \(\approx\) \(0.01427318236 - 0.7019561792i\)
\(L(1)\) \(\approx\) \(0.8259433256 - 0.3873358325i\)
\(L(1)\) \(\approx\) \(0.8259433256 - 0.3873358325i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.433 - 0.900i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (-0.781 - 0.623i)T \)
17 \( 1 + (-0.974 - 0.222i)T \)
19 \( 1 + T \)
23 \( 1 + (0.974 - 0.222i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (-0.974 + 0.222i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.781 - 0.623i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.781 + 0.623i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
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

−21.894846521870743607574163312012, −21.44810226622733371560838261920, −20.54143408746355055450486301542, −19.848606435472681917184796646450, −19.13080692700180545577864566120, −18.23253945786799847079961510517, −17.20780976156504208647842869100, −16.43859709107533871351083568234, −15.817849238853573615922468150107, −15.02290402902285758870765174299, −14.220546709375800785413419811917, −13.54315366834009566846079519133, −12.61653295574317993069512413876, −11.34587205846091268758330928348, −10.93484764465177051341852283178, −9.88653207690440132882952072360, −9.19041544277586844459124159284, −8.44664879802366419195403955958, −7.51153602137922281264293567747, −6.48842852821461844537483529079, −5.18061405536978198942273931508, −4.81252837617375343290552340462, −3.50139865303961159938912222570, −2.90573036468199474971814764689, −1.710536335490329897328244978780, 0.25949861873980095459302326267, 1.717519235069363044269420340266, 2.54216112052513814938515024943, 3.399967028691282023142455714179, 4.78542713372068153956260420420, 5.55912815250535029566103916561, 6.84309495002037389841764840892, 7.306778632941233821170046556275, 8.15069517052548182410946991363, 9.083303871110846638456630606242, 9.89573126026299144141944851628, 10.9216994319763963537983399150, 11.94969005545903750920502757616, 12.58489297878315999103324306673, 13.34745536384759322029525295125, 14.013369485609932563692193644994, 15.124912587597702421274308275452, 15.44093878908571955314281538699, 16.81274607860971199193708218675, 17.61855896545954488336485542483, 18.160133451607754485939220305174, 18.94406071032798188451999058191, 19.88888756832308252006337492755, 20.303131272554058225690692789430, 21.12185035020088426856257381029

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