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Properties

Label 2-65-13.3-c3-0-12
Degree $2$
Conductor $65$
Sign $-0.997 + 0.0764i$
Analytic cond. $3.83512$
Root an. cond. $1.95834$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.272 − 0.472i)2-s + (−3.51 − 6.08i)3-s + (3.85 − 6.67i)4-s − 5·5-s + (−1.91 + 3.31i)6-s + (−11.4 + 19.8i)7-s − 8.56·8-s + (−11.1 + 19.3i)9-s + (1.36 + 2.36i)10-s + (−4.27 − 7.40i)11-s − 54.0·12-s + (−15.2 − 44.3i)13-s + 12.4·14-s + (17.5 + 30.4i)15-s + (−28.4 − 49.3i)16-s + (−16.3 + 28.3i)17-s + ⋯
L(s)  = 1  + (−0.0964 − 0.166i)2-s + (−0.675 − 1.17i)3-s + (0.481 − 0.833i)4-s − 0.447·5-s + (−0.130 + 0.225i)6-s + (−0.617 + 1.06i)7-s − 0.378·8-s + (−0.413 + 0.715i)9-s + (0.0431 + 0.0746i)10-s + (−0.117 − 0.203i)11-s − 1.30·12-s + (−0.325 − 0.945i)13-s + 0.238·14-s + (0.302 + 0.523i)15-s + (−0.444 − 0.770i)16-s + (−0.233 + 0.403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0764i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.997 + 0.0764i$
Analytic conductor: \(3.83512\)
Root analytic conductor: \(1.95834\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :3/2),\ -0.997 + 0.0764i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0279264 - 0.729242i\)
\(L(\frac12)\) \(\approx\) \(0.0279264 - 0.729242i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
13 \( 1 + (15.2 + 44.3i)T \)
good2 \( 1 + (0.272 + 0.472i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (3.51 + 6.08i)T + (-13.5 + 23.3i)T^{2} \)
7 \( 1 + (11.4 - 19.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (4.27 + 7.40i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (16.3 - 28.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-43.5 + 75.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-4.11 - 7.13i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (107. + 185. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 279.T + 2.97e4T^{2} \)
37 \( 1 + (136. + 237. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-42.9 - 74.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-190. + 329. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 435.T + 1.03e5T^{2} \)
53 \( 1 + 436.T + 1.48e5T^{2} \)
59 \( 1 + (374. - 648. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-35.3 + 61.2i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (103. + 178. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (91.2 - 158. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 188.T + 3.89e5T^{2} \)
79 \( 1 - 923.T + 4.93e5T^{2} \)
83 \( 1 - 1.21e3T + 5.71e5T^{2} \)
89 \( 1 + (-9.88 - 17.1i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (270. - 469. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.60213349078851693639726746309, −12.43741239657949612273232516251, −11.80645474330971909935007794477, −10.69420161142212731742157036751, −9.246963947152889682107191021802, −7.59250318146220923299561837164, −6.35408632554801564522122891093, −5.51752276324304326632289273687, −2.51688386841147950162303679601, −0.53491163176965653214017056318, 3.50696408603917670402067258875, 4.60180246812500858918098474112, 6.56762095488796972559222249273, 7.68281816604320303944617129061, 9.374432829255753456510042116360, 10.47446927647007492051559144786, 11.44975134452266797485424846276, 12.43414430682304353788021259147, 13.89596966789381950969673052489, 15.37936767079897681484610680362

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