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Properties

Label 2-65-65.12-c2-0-6
Degree $2$
Conductor $65$
Sign $0.996 - 0.0876i$
Analytic cond. $1.77112$
Root an. cond. $1.33083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.14 + 2.14i)2-s + (1.85 − 1.85i)3-s − 5.18i·4-s + (1.93 − 4.60i)5-s + 7.96i·6-s + (5.84 − 5.84i)7-s + (2.53 + 2.53i)8-s + 2.09i·9-s + (5.72 + 14.0i)10-s + 8.13i·11-s + (−9.62 − 9.62i)12-s + (12.9 + 0.884i)13-s + 25.0i·14-s + (−4.96 − 12.1i)15-s + 9.88·16-s + (−15.0 − 15.0i)17-s + ⋯
L(s)  = 1  + (−1.07 + 1.07i)2-s + (0.619 − 0.619i)3-s − 1.29i·4-s + (0.387 − 0.921i)5-s + 1.32i·6-s + (0.835 − 0.835i)7-s + (0.316 + 0.316i)8-s + 0.232i·9-s + (0.572 + 1.40i)10-s + 0.739i·11-s + (−0.802 − 0.802i)12-s + (0.997 + 0.0680i)13-s + 1.79i·14-s + (−0.331 − 0.810i)15-s + 0.617·16-s + (−0.887 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0876i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.996 - 0.0876i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ 0.996 - 0.0876i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.949367 + 0.0416951i\)
\(L(\frac12)\) \(\approx\) \(0.949367 + 0.0416951i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.93 + 4.60i)T \)
13 \( 1 + (-12.9 - 0.884i)T \)
good2 \( 1 + (2.14 - 2.14i)T - 4iT^{2} \)
3 \( 1 + (-1.85 + 1.85i)T - 9iT^{2} \)
7 \( 1 + (-5.84 + 5.84i)T - 49iT^{2} \)
11 \( 1 - 8.13iT - 121T^{2} \)
17 \( 1 + (15.0 + 15.0i)T + 289iT^{2} \)
19 \( 1 + 7.91T + 361T^{2} \)
23 \( 1 + (0.359 - 0.359i)T - 529iT^{2} \)
29 \( 1 - 39.6iT - 841T^{2} \)
31 \( 1 + 10.3iT - 961T^{2} \)
37 \( 1 + (-14.3 + 14.3i)T - 1.36e3iT^{2} \)
41 \( 1 - 38.9iT - 1.68e3T^{2} \)
43 \( 1 + (37.8 - 37.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (56.8 - 56.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (-49.5 + 49.5i)T - 2.80e3iT^{2} \)
59 \( 1 + 37.9T + 3.48e3T^{2} \)
61 \( 1 + 27.8T + 3.72e3T^{2} \)
67 \( 1 + (-17.5 + 17.5i)T - 4.48e3iT^{2} \)
71 \( 1 + 106. iT - 5.04e3T^{2} \)
73 \( 1 + (-67.9 - 67.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 106. iT - 6.24e3T^{2} \)
83 \( 1 + (8.61 + 8.61i)T + 6.88e3iT^{2} \)
89 \( 1 - 70.4T + 7.92e3T^{2} \)
97 \( 1 + (33.7 - 33.7i)T - 9.40e3iT^{2} \)
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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

−14.74044570169815539547595799107, −13.72704227221933636813917974937, −12.82333978148154664592828367558, −10.97440145468620899472138072858, −9.523075428742676657743711240657, −8.504426791243964159486708866728, −7.77278535989665101064524860253, −6.65816871109121769008667391066, −4.82283613455338510459434832803, −1.44399971035823897045923149109, 2.17209290421850088557753540509, 3.57584545806894890591136267642, 6.09485631212718637031926897909, 8.351986800726478263798800955903, 8.875497170529723886365882926456, 10.14050323385667161816518229420, 10.98157849678348673906075910681, 11.83036108192820823451558119923, 13.54083494834219919617119012907, 14.81917521098202908018376949589

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