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Properties

Label 1-2900-2900.1519-r0-0-0
Degree $1$
Conductor $2900$
Sign $0.120 - 0.992i$
Analytic cond. $13.4675$
Root an. cond. $13.4675$
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  + (−0.0896 + 0.995i)3-s + (−0.222 − 0.974i)7-s + (−0.983 − 0.178i)9-s + (0.178 + 0.983i)11-s + (0.473 + 0.880i)13-s + (−0.951 + 0.309i)17-s + (−0.0896 − 0.995i)19-s + (0.990 − 0.134i)21-s + (−0.0448 + 0.998i)23-s + (0.266 − 0.963i)27-s + (−0.834 − 0.550i)31-s + (−0.995 + 0.0896i)33-s + (0.178 − 0.983i)37-s + (−0.919 + 0.393i)39-s + (0.587 − 0.809i)41-s + ⋯
L(s)  = 1  + (−0.0896 + 0.995i)3-s + (−0.222 − 0.974i)7-s + (−0.983 − 0.178i)9-s + (0.178 + 0.983i)11-s + (0.473 + 0.880i)13-s + (−0.951 + 0.309i)17-s + (−0.0896 − 0.995i)19-s + (0.990 − 0.134i)21-s + (−0.0448 + 0.998i)23-s + (0.266 − 0.963i)27-s + (−0.834 − 0.550i)31-s + (−0.995 + 0.0896i)33-s + (0.178 − 0.983i)37-s + (−0.919 + 0.393i)39-s + (0.587 − 0.809i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.120 - 0.992i$
Analytic conductor: \(13.4675\)
Root analytic conductor: \(13.4675\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2900,\ (0:\ ),\ 0.120 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3655912961 - 0.3238102793i\)
\(L(\frac12)\) \(\approx\) \(0.3655912961 - 0.3238102793i\)
\(L(1)\) \(\approx\) \(0.7948377165 + 0.1791956235i\)
\(L(1)\) \(\approx\) \(0.7948377165 + 0.1791956235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.0896 + 0.995i)T \)
7 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (0.178 + 0.983i)T \)
13 \( 1 + (0.473 + 0.880i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.0896 - 0.995i)T \)
23 \( 1 + (-0.0448 + 0.998i)T \)
31 \( 1 + (-0.834 - 0.550i)T \)
37 \( 1 + (0.178 - 0.983i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (-0.781 - 0.623i)T \)
47 \( 1 + (-0.722 - 0.691i)T \)
53 \( 1 + (-0.936 + 0.351i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.657 - 0.753i)T \)
67 \( 1 + (0.691 + 0.722i)T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (0.998 + 0.0448i)T \)
79 \( 1 + (-0.990 + 0.134i)T \)
83 \( 1 + (-0.995 + 0.0896i)T \)
89 \( 1 + (-0.266 + 0.963i)T \)
97 \( 1 + (0.512 - 0.858i)T \)
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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

−19.1589147847803699629373538316, −18.47014642113235612190232751564, −18.17246712558118137797228073239, −17.36755976799216038859280426539, −16.39008180170140414142671331568, −16.00134093579477606490193882708, −14.906736600332458076417122855632, −14.39699329466845679043544635890, −13.483302158154145926237950138954, −12.87414956967132686514630357965, −12.39661805651760035271794762327, −11.428818582991731819397717599666, −11.08299142875807460827128648238, −10.00720837285197961601376796126, −9.00056125206912498481179755629, −8.35009023620178318968187727120, −7.99059244555643688964778600273, −6.73894768102161311549630422454, −6.22470544698947764065088538929, −5.655638583629812966906337644458, −4.77978302926557401716279017399, −3.40432234632238219795946173027, −2.8700098817240946038839834869, −1.96188081133657245050960457497, −1.04301809632534406015264828597, 0.16084740550486523490277540674, 1.60825192786807033518317999964, 2.521271171202474471212297364934, 3.80142874617475571227205455788, 4.03084501971839879549966615078, 4.833303201420041307927671358214, 5.700889149885873215601743383566, 6.772339071291513131397253199039, 7.14940509326136154476650643783, 8.30950491082619608025626732536, 9.18703946629110135768631904047, 9.59041961604692060657654443122, 10.402789526462801938877395359417, 11.178083382087962447126908516279, 11.50797329398633193811082494451, 12.70871843797477968819852472912, 13.40299334756970828581032113616, 14.1254506920612893381201235667, 14.822999683028062633845244620043, 15.59731325065394903620358684581, 16.06463754726371712495338127256, 16.94013867729939435351004534779, 17.40765996351997545570691841154, 18.01496562388665989590738473332, 19.22255922151150477094339741851

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