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Properties

Label 2-65-65.29-c1-0-3
Degree $2$
Conductor $65$
Sign $0.939 + 0.341i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.02 − 0.593i)2-s + (−0.298 + 0.172i)3-s + (−0.295 + 0.511i)4-s + (1.44 − 1.71i)5-s + (−0.204 + 0.354i)6-s + (−1.75 − 1.01i)7-s + 3.07i·8-s + (−1.44 + 2.49i)9-s + (0.465 − 2.61i)10-s + (−1.94 − 3.36i)11-s − 0.203i·12-s + (−2.96 + 2.05i)13-s − 2.40·14-s + (−0.135 + 0.759i)15-s + (1.23 + 2.14i)16-s + (4.71 + 2.72i)17-s + ⋯
L(s)  = 1  + (0.727 − 0.419i)2-s + (−0.172 + 0.0996i)3-s + (−0.147 + 0.255i)4-s + (0.644 − 0.764i)5-s + (−0.0836 + 0.144i)6-s + (−0.664 − 0.383i)7-s + 1.08i·8-s + (−0.480 + 0.831i)9-s + (0.147 − 0.826i)10-s + (−0.585 − 1.01i)11-s − 0.0588i·12-s + (−0.821 + 0.570i)13-s − 0.644·14-s + (−0.0349 + 0.196i)15-s + (0.308 + 0.535i)16-s + (1.14 + 0.660i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.939 + 0.341i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.939 + 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10261 - 0.194248i\)
\(L(\frac12)\) \(\approx\) \(1.10261 - 0.194248i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.44 + 1.71i)T \)
13 \( 1 + (2.96 - 2.05i)T \)
good2 \( 1 + (-1.02 + 0.593i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.298 - 0.172i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.75 + 1.01i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.94 + 3.36i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.71 - 2.72i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.94 + 5.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.298 + 0.172i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.18T + 31T^{2} \)
37 \( 1 + (-4.71 + 2.72i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0902 + 0.156i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.15 + 0.669i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.2iT - 47T^{2} \)
53 \( 1 + 2.42iT - 53T^{2} \)
59 \( 1 + (3.53 - 6.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 - 5.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.81 - 2.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.940 + 1.62i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.86iT - 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 7.83iT - 83T^{2} \)
89 \( 1 + (-6.12 - 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.02 - 2.90i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.31776187829125190067198215703, −13.55529705813391965643564176531, −12.85228949303170377662839101562, −11.72085225862550433166169966647, −10.48387240336596857544426063796, −9.107100402598138105903778073546, −7.84869407634030784682365785594, −5.75786958389092996629339701671, −4.73381412011506811017839262963, −2.88873652432080843097933219345, 3.14663487049026575016803075665, 5.28749380845141342223769014597, 6.18016441063784788648882393233, 7.38673476782081625125459716214, 9.728107143353509843548645390698, 9.997002638537442740953613316108, 12.01775460414634166974312981584, 12.85798499857755303223149771144, 14.11159266131250887876339887908, 14.81843884736234023738329320475

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