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Properties

Label 2-1519-1519.309-c0-0-0
Degree $2$
Conductor $1519$
Sign $0.0320 + 0.999i$
Analytic cond. $0.758079$
Root an. cond. $0.870677$
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  + (−0.277 + 1.21i)2-s + (−0.499 − 0.240i)4-s + (−1.12 + 1.40i)5-s + (−0.900 + 0.433i)7-s + (−0.346 + 0.433i)8-s + (−0.222 − 0.974i)9-s + (−1.40 − 1.75i)10-s + (−0.277 − 1.21i)14-s + (−0.777 − 0.974i)16-s + 1.24·18-s − 0.445·19-s + (0.900 − 0.433i)20-s + (−0.500 − 2.19i)25-s + 0.554·28-s + 31-s + (0.900 − 0.433i)32-s + ⋯
L(s)  = 1  + (−0.277 + 1.21i)2-s + (−0.499 − 0.240i)4-s + (−1.12 + 1.40i)5-s + (−0.900 + 0.433i)7-s + (−0.346 + 0.433i)8-s + (−0.222 − 0.974i)9-s + (−1.40 − 1.75i)10-s + (−0.277 − 1.21i)14-s + (−0.777 − 0.974i)16-s + 1.24·18-s − 0.445·19-s + (0.900 − 0.433i)20-s + (−0.500 − 2.19i)25-s + 0.554·28-s + 31-s + (0.900 − 0.433i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1519\)    =    \(7^{2} \cdot 31\)
Sign: $0.0320 + 0.999i$
Analytic conductor: \(0.758079\)
Root analytic conductor: \(0.870677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1519} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1519,\ (\ :0),\ 0.0320 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2508798820\)
\(L(\frac12)\) \(\approx\) \(0.2508798820\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 - T \)
good2 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
3 \( 1 + (0.222 + 0.974i)T^{2} \)
5 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.623 + 0.781i)T^{2} \)
19 \( 1 + 0.445T + T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
37 \( 1 + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
61 \( 1 + (-0.623 + 0.781i)T^{2} \)
67 \( 1 + 1.80T + T^{2} \)
71 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.900 - 0.433i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
97 \( 1 + 1.80T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.08444418028938293722033921293, −9.265427203695988500447235862021, −8.410952759558643908612333514455, −7.75211064756502330448468672830, −6.86844811733838053406540334381, −6.48363389929439010368497259784, −5.90482763861753971311223765191, −4.43560013598680494689375457822, −3.26856588300797237163265212973, −2.79732530292717687149745714034, 0.21530720058150475169828096201, 1.54399546224177638171299728739, 2.85547745721104318162497826015, 3.84417509347823949468343064287, 4.49197038225466249963326872547, 5.52679684225484258930086703594, 6.76948277043785216732539533439, 7.69307000523007098506316977400, 8.576915612871176808472237410573, 9.029582460667286212339079816931

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