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Properties

Label 1-1519-1519.284-r0-0-0
Degree $1$
Conductor $1519$
Sign $0.863 - 0.504i$
Analytic cond. $7.05420$
Root an. cond. $7.05420$
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  + (0.955 + 0.294i)2-s + (−0.988 + 0.149i)3-s + (0.826 + 0.563i)4-s + (0.365 − 0.930i)5-s + (−0.988 − 0.149i)6-s + (0.623 + 0.781i)8-s + (0.955 − 0.294i)9-s + (0.623 − 0.781i)10-s + (−0.222 − 0.974i)11-s + (−0.900 − 0.433i)12-s + (0.955 + 0.294i)13-s + (−0.222 + 0.974i)15-s + (0.365 + 0.930i)16-s + (0.826 − 0.563i)17-s + 18-s + 19-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)2-s + (−0.988 + 0.149i)3-s + (0.826 + 0.563i)4-s + (0.365 − 0.930i)5-s + (−0.988 − 0.149i)6-s + (0.623 + 0.781i)8-s + (0.955 − 0.294i)9-s + (0.623 − 0.781i)10-s + (−0.222 − 0.974i)11-s + (−0.900 − 0.433i)12-s + (0.955 + 0.294i)13-s + (−0.222 + 0.974i)15-s + (0.365 + 0.930i)16-s + (0.826 − 0.563i)17-s + 18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1519\)    =    \(7^{2} \cdot 31\)
Sign: $0.863 - 0.504i$
Analytic conductor: \(7.05420\)
Root analytic conductor: \(7.05420\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1519} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1519,\ (0:\ ),\ 0.863 - 0.504i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.390915204 - 0.6480224245i\)
\(L(\frac12)\) \(\approx\) \(2.390915204 - 0.6480224245i\)
\(L(1)\) \(\approx\) \(1.625945232 - 0.04971926902i\)
\(L(1)\) \(\approx\) \(1.625945232 - 0.04971926902i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.955 + 0.294i)T \)
3 \( 1 + (-0.988 + 0.149i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (0.955 + 0.294i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
19 \( 1 + T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (-0.988 + 0.149i)T \)
43 \( 1 + (-0.988 - 0.149i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (0.365 + 0.930i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + T \)
71 \( 1 + (0.0747 - 0.997i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.733 - 0.680i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
97 \( 1 + T \)
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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

−20.8748586924380622304006950954, −20.13873551596273299069277161604, −18.96876806236075375761642279458, −18.52719904831856002051800253245, −17.70797354447063998626501580404, −16.953701768075179832628845903430, −15.89913299197264141273602698031, −15.38443961702647768719440958254, −14.61586604746960125323218161151, −13.6611990707560242754524216003, −13.119878722385237574583975192870, −12.25141446948130928049814302280, −11.55590500700981293700083221231, −10.89926367115041842532264828032, −10.16151078879716698342942996253, −9.67348864706249253389520327323, −7.83970575725995284535556510800, −7.06939275180364676626519011341, −6.49813473648764724228412759660, −5.46629053026634987061861962963, −5.25685119069896356178722837207, −3.836204598574432169522401573565, −3.29150971442349215742496047053, −1.952310944252696654227456027765, −1.32925973176749816613535119290, 0.81032668125127534425318270838, 1.78366424633882795905082579251, 3.2178627340507768378439899131, 4.01649325320264989760413533571, 4.98431738181830180395915027995, 5.533679992730495739952357160312, 6.09399922946956307896427069633, 7.00640082309309676286743386116, 8.01170203109527239097048835366, 8.87704549842099457324285541349, 9.94632896539019094791567283047, 10.8670690313441978843675744596, 11.606908200609379380000282365052, 12.1697492528278035284655758128, 13.011710045342181709108044663938, 13.6174830830490092004965977787, 14.29613818842856538430361202869, 15.62011063698726857909218418197, 16.01965733515693714496325700012, 16.69148484506787344479020785963, 17.09787880790703709773352505173, 18.23742050219438900100927993661, 18.85559146617447725789812979973, 20.25334623684522686489866749992, 20.82118761708779928004469460815

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