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Properties

Label 1-1976-1976.435-r0-0-0
Degree $1$
Conductor $1976$
Sign $-0.891 - 0.452i$
Analytic cond. $9.17650$
Root an. cond. $9.17650$
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  + (0.766 − 0.642i)3-s + (0.642 + 0.766i)5-s + (−0.866 + 0.5i)7-s + (0.173 − 0.984i)9-s + (−0.866 − 0.5i)11-s + (0.984 + 0.173i)15-s + (−0.766 + 0.642i)17-s + (−0.342 + 0.939i)21-s + (0.173 − 0.984i)23-s + (−0.173 + 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.766 − 0.642i)29-s + (−0.866 − 0.5i)31-s + (−0.984 + 0.173i)33-s + (−0.939 − 0.342i)35-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (0.642 + 0.766i)5-s + (−0.866 + 0.5i)7-s + (0.173 − 0.984i)9-s + (−0.866 − 0.5i)11-s + (0.984 + 0.173i)15-s + (−0.766 + 0.642i)17-s + (−0.342 + 0.939i)21-s + (0.173 − 0.984i)23-s + (−0.173 + 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.766 − 0.642i)29-s + (−0.866 − 0.5i)31-s + (−0.984 + 0.173i)33-s + (−0.939 − 0.342i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1976\)    =    \(2^{3} \cdot 13 \cdot 19\)
Sign: $-0.891 - 0.452i$
Analytic conductor: \(9.17650\)
Root analytic conductor: \(9.17650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1976} (435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1976,\ (0:\ ),\ -0.891 - 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1527338023 - 0.6390805034i\)
\(L(\frac12)\) \(\approx\) \(0.1527338023 - 0.6390805034i\)
\(L(1)\) \(\approx\) \(1.013463320 - 0.1868470455i\)
\(L(1)\) \(\approx\) \(1.013463320 - 0.1868470455i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (0.642 + 0.766i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (-0.984 + 0.173i)T \)
73 \( 1 + (-0.984 + 0.173i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (-0.342 - 0.939i)T \)
97 \( 1 + (-0.642 - 0.766i)T \)
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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

−20.36351046688304511161352463145, −19.82765410360357106761172507114, −19.00045894583937433651047317424, −18.04074391567292876020004673209, −17.32594539458865762970827157534, −16.338505612530724189105750577226, −16.075818799756515079943800205213, −15.271455793616433639576980661074, −14.38553042079150675022131916393, −13.46380908743272530835702113722, −13.218890870511548630695414100212, −12.47832492612524921798628810255, −11.19488420886174910661790356016, −10.3822755748779088640369724348, −9.71648819905360354999438226188, −9.23496988042302620125559793280, −8.48230485177506202799447621388, −7.49490481534498682604847792080, −6.82207470588604265940049135324, −5.51429359030194782232915855111, −5.01014495908037624020637297700, −4.09072492319648178557049179912, −3.24096094107061705365061326258, −2.37638648334021988346623959151, −1.47848181991163224608547375364, 0.17594784242743019999967947906, 1.84198483294427296777070980930, 2.42114297167507147544679469726, 3.14111317664670622720516111182, 3.90385637477643859821674246479, 5.406645611296201591259185573920, 6.139511875927764434688860553570, 6.749117238738325240450040877, 7.50611914939452110276744185262, 8.51916972799253353544720580948, 9.05400085136079299246832708882, 9.96249384054103646284698319939, 10.59419958366136548191488385773, 11.5478533106975631325226403702, 12.621571728094757559514585157858, 13.1179675428272023942689507374, 13.66442549641632756178489074553, 14.509437561572752352838197840964, 15.22129865824289766781403915576, 15.76758885591781255369169466330, 16.874179176189613756392976207876, 17.69066510499952408062700746070, 18.46685059143494709140684243388, 18.922851190383583656777009816655, 19.352297282144286713620347104919

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