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Properties

Label 2-6-3.2-c30-0-2
Degree $2$
Conductor $6$
Sign $0.802 + 0.597i$
Analytic cond. $34.2085$
Root an. cond. $5.84880$
Motivic weight $30$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.31e4i·2-s + (−8.56e6 + 1.15e7i)3-s − 5.36e8·4-s + 4.58e10i·5-s + (−2.66e11 − 1.98e11i)6-s − 5.02e12·7-s − 1.24e13i·8-s + (−5.90e13 − 1.97e14i)9-s − 1.06e15·10-s + 5.42e15i·11-s + (4.59e15 − 6.17e15i)12-s − 6.28e16·13-s − 1.16e17i·14-s + (−5.28e17 − 3.93e17i)15-s + 2.88e17·16-s + 1.97e18i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.597 + 0.802i)3-s − 0.500·4-s + 1.50i·5-s + (−0.567 − 0.422i)6-s − 1.05·7-s − 0.353i·8-s + (−0.287 − 0.957i)9-s − 1.06·10-s + 1.29i·11-s + (0.298 − 0.401i)12-s − 1.22·13-s − 0.748i·14-s + (−1.20 − 0.897i)15-s + 0.250·16-s + 0.689i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 + 0.597i)\, \overline{\Lambda}(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & (0.802 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.802 + 0.597i$
Analytic conductor: \(34.2085\)
Root analytic conductor: \(5.84880\)
Motivic weight: \(30\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :15),\ 0.802 + 0.597i)\)

Particular Values

\(L(\frac{31}{2})\) \(\approx\) \(0.4509084714\)
\(L(\frac12)\) \(\approx\) \(0.4509084714\)
\(L(16)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.31e4iT \)
3 \( 1 + (8.56e6 - 1.15e7i)T \)
good5 \( 1 - 4.58e10iT - 9.31e20T^{2} \)
7 \( 1 + 5.02e12T + 2.25e25T^{2} \)
11 \( 1 - 5.42e15iT - 1.74e31T^{2} \)
13 \( 1 + 6.28e16T + 2.61e33T^{2} \)
17 \( 1 - 1.97e18iT - 8.19e36T^{2} \)
19 \( 1 + 1.24e19T + 2.30e38T^{2} \)
23 \( 1 - 1.89e19iT - 7.10e40T^{2} \)
29 \( 1 - 1.35e22iT - 7.44e43T^{2} \)
31 \( 1 - 3.24e22T + 5.50e44T^{2} \)
37 \( 1 - 6.42e23T + 1.11e47T^{2} \)
41 \( 1 + 5.25e23iT - 2.41e48T^{2} \)
43 \( 1 - 5.90e23T + 1.00e49T^{2} \)
47 \( 1 - 3.38e22iT - 1.45e50T^{2} \)
53 \( 1 - 7.47e25iT - 5.34e51T^{2} \)
59 \( 1 - 2.93e26iT - 1.33e53T^{2} \)
61 \( 1 + 3.02e26T + 3.62e53T^{2} \)
67 \( 1 + 3.18e27T + 6.05e54T^{2} \)
71 \( 1 + 2.30e27iT - 3.44e55T^{2} \)
73 \( 1 - 1.38e28T + 7.93e55T^{2} \)
79 \( 1 + 3.59e28T + 8.48e56T^{2} \)
83 \( 1 + 2.29e28iT - 3.73e57T^{2} \)
89 \( 1 - 1.44e29iT - 3.03e58T^{2} \)
97 \( 1 + 6.73e29T + 4.01e59T^{2} \)
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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

−16.87399902863484024061037953545, −15.31602964460495246364127836988, −14.66647692802163998787575725285, −12.45297489796661754263283957950, −10.49764285555316578109234954362, −9.635184776094845824473973828803, −7.12621875555914569923527748247, −6.20729654286323420033058414322, −4.40151143398295059417145939838, −2.86250269701745799081290968768, 0.20932013313271428401353253124, 0.802346225829123791320482572508, 2.55788912780043908623306979683, 4.65660381159677329732290204931, 6.07528415699371800033448337013, 8.181654995253283875193659324628, 9.672461588707804822335312857668, 11.62133323091986234018238035226, 12.69220454929202515769840189531, 13.51696878593497610693810899105

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