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Properties

Label 1-3100-3100.2187-r1-0-0
Degree $1$
Conductor $3100$
Sign $0.308 - 0.951i$
Analytic cond. $333.141$
Root an. cond. $333.141$
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  + (0.743 − 0.669i)3-s + (−0.207 + 0.978i)7-s + (0.104 − 0.994i)9-s + (0.913 + 0.406i)11-s + (0.743 + 0.669i)13-s + (−0.994 + 0.104i)17-s + (−0.978 − 0.207i)19-s + (0.5 + 0.866i)21-s − i·23-s + (−0.587 − 0.809i)27-s + 29-s + (0.951 − 0.309i)33-s + (0.743 − 0.669i)37-s + 39-s + (0.913 + 0.406i)41-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)3-s + (−0.207 + 0.978i)7-s + (0.104 − 0.994i)9-s + (0.913 + 0.406i)11-s + (0.743 + 0.669i)13-s + (−0.994 + 0.104i)17-s + (−0.978 − 0.207i)19-s + (0.5 + 0.866i)21-s − i·23-s + (−0.587 − 0.809i)27-s + 29-s + (0.951 − 0.309i)33-s + (0.743 − 0.669i)37-s + 39-s + (0.913 + 0.406i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $0.308 - 0.951i$
Analytic conductor: \(333.141\)
Root analytic conductor: \(333.141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3100} (2187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3100,\ (1:\ ),\ 0.308 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.282417097 - 1.659952426i\)
\(L(\frac12)\) \(\approx\) \(2.282417097 - 1.659952426i\)
\(L(1)\) \(\approx\) \(1.358181516 - 0.2511653107i\)
\(L(1)\) \(\approx\) \(1.358181516 - 0.2511653107i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.743 - 0.669i)T \)
7 \( 1 + (-0.207 + 0.978i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (0.743 + 0.669i)T \)
17 \( 1 + (-0.994 + 0.104i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 - iT \)
29 \( 1 + T \)
37 \( 1 + (0.743 - 0.669i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.406 - 0.913i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.743 + 0.669i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.994 + 0.104i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (-0.406 + 0.913i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−19.42790175963359335539184719738, −18.232691597194191063524835051566, −17.469604404538365352623226202533, −16.774674305580874733805039026399, −16.13534029525680114510440579316, −15.49172369545861959048344731775, −14.745658551231826790974834304168, −14.10387031222570298858959840342, −13.341033158267683810140975115522, −13.06561989105695589499581012929, −11.739964541598929376583848613, −10.96297462969822267773125114722, −10.49729357483043474674132264756, −9.677086079794314751183236635530, −9.002771299533497345167336441954, −8.290470816830722095179722954376, −7.65622785685790503649195398975, −6.63686931655198721380218362047, −6.04946817092464815912613075806, −4.81201385760125259166648401715, −4.18431646854000709777470533707, −3.55552305274787625022080807418, −2.852213653850406088244704624872, −1.71357929250142642060863372244, −0.831594184035083107035365041280, 0.44473895539714066919065140913, 1.60292705164591484487369088384, 2.20803255509543537904491412622, 2.92393363637652332714907714803, 4.0160349859275084460653430539, 4.533831602797116334351994914, 5.94199843949108605710906970283, 6.52178350795886702286871037595, 6.92928824921838689255400964651, 8.1574481241575267174545703590, 8.761189633719721633541082793209, 9.08694135251605799727503332516, 9.96272160566245135988674915630, 11.09705380686304810107756859332, 11.72659456057695198340315100589, 12.57504701482012915539219993054, 12.89338791081794452780339033343, 13.886627508803277759499291176484, 14.432517848072179567634792947533, 15.14977915193912158725299264720, 15.69052884838544406500794989650, 16.616141200134279311199861734691, 17.465106472268874200332154285048, 18.189233861539943955576328866, 18.6585957120504633255641809533

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