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Properties

Label 1-5915-5915.1223-r0-0-0
Degree $1$
Conductor $5915$
Sign $-0.338 - 0.940i$
Analytic cond. $27.4691$
Root an. cond. $27.4691$
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  + (−0.999 + 0.0402i)2-s + (−0.721 − 0.692i)3-s + (0.996 − 0.0804i)4-s + (0.748 + 0.663i)6-s + (−0.992 + 0.120i)8-s + (0.0402 + 0.999i)9-s + (−0.0402 + 0.999i)11-s + (−0.774 − 0.632i)12-s + (0.987 − 0.160i)16-s + (0.391 − 0.919i)17-s + (−0.0804 − 0.996i)18-s + (−0.5 − 0.866i)19-s − i·22-s + (0.866 − 0.5i)23-s + (0.799 + 0.600i)24-s + ⋯
L(s)  = 1  + (−0.999 + 0.0402i)2-s + (−0.721 − 0.692i)3-s + (0.996 − 0.0804i)4-s + (0.748 + 0.663i)6-s + (−0.992 + 0.120i)8-s + (0.0402 + 0.999i)9-s + (−0.0402 + 0.999i)11-s + (−0.774 − 0.632i)12-s + (0.987 − 0.160i)16-s + (0.391 − 0.919i)17-s + (−0.0804 − 0.996i)18-s + (−0.5 − 0.866i)19-s − i·22-s + (0.866 − 0.5i)23-s + (0.799 + 0.600i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5915\)    =    \(5 \cdot 7 \cdot 13^{2}\)
Sign: $-0.338 - 0.940i$
Analytic conductor: \(27.4691\)
Root analytic conductor: \(27.4691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5915} (1223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5915,\ (0:\ ),\ -0.338 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3753309297 - 0.5341225399i\)
\(L(\frac12)\) \(\approx\) \(0.3753309297 - 0.5341225399i\)
\(L(1)\) \(\approx\) \(0.5348687818 - 0.1450677032i\)
\(L(1)\) \(\approx\) \(0.5348687818 - 0.1450677032i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.999 + 0.0402i)T \)
3 \( 1 + (-0.721 - 0.692i)T \)
11 \( 1 + (-0.0402 + 0.999i)T \)
17 \( 1 + (0.391 - 0.919i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.885 - 0.464i)T \)
31 \( 1 + (-0.948 + 0.316i)T \)
37 \( 1 + (0.979 + 0.200i)T \)
41 \( 1 + (0.970 + 0.239i)T \)
43 \( 1 + (-0.663 - 0.748i)T \)
47 \( 1 + (0.0804 - 0.996i)T \)
53 \( 1 + (0.391 - 0.919i)T \)
59 \( 1 + (0.987 + 0.160i)T \)
61 \( 1 + (0.919 - 0.391i)T \)
67 \( 1 + (0.903 - 0.428i)T \)
71 \( 1 + (-0.970 - 0.239i)T \)
73 \( 1 + (0.999 + 0.0402i)T \)
79 \( 1 + (0.996 + 0.0804i)T \)
83 \( 1 + (0.239 + 0.970i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.935 - 0.354i)T \)
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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

−17.87789250315633004279114590450, −17.17737674947420535748024397566, −16.50820073292478773275700593410, −16.44249139111006015830038128124, −15.49477678202057535232053030379, −14.834425621531122431996315279025, −14.397080751507193664785515642185, −12.98213336873568601407374318523, −12.60999427715399314607209578755, −11.641941698978296051975130289078, −11.09743703482078497772529659062, −10.73455379870272196935991930466, −9.973021019008654948841237892494, −9.329603555445731966404876538302, −8.74140197713352043035307523043, −7.986204165688151097644885242943, −7.26792891897310851541100252647, −6.32225143766138861850367212603, −5.853216604499680681690878577267, −5.29535641790483614002496888878, −4.00135974219553338401840521191, −3.55482209640090097632778164796, −2.65486659488565368230000798187, −1.528663461738874441261470446105, −0.83008660869343302168638719982, 0.363333067337695732926874323949, 1.08871721024561914522286424643, 2.18377648300762737167215324220, 2.41601869046011948353069595035, 3.66689856111857261910272421947, 4.87277489010664013372781703855, 5.34899377068535802892957456200, 6.29453835703498947395448364517, 7.01368073411705465305401335178, 7.25879608218401064443439133456, 8.061992710044722778488803018249, 8.85319481919361743795256888838, 9.59632871896045431767203756959, 10.18553360168568235296537246954, 11.05995782663567920891576941212, 11.38658242349399310502520115940, 12.152974231980267466272117420926, 12.79991584352237055567839793207, 13.33189516023543104281693454440, 14.465556213813177488725463412, 15.044041979011992303935472801026, 15.77281322470490009967509018487, 16.58931659400726131580953832497, 16.90447557365141232195450672628, 17.64980352362870982867724772267

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