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Properties

Label 2-65-65.4-c1-0-0
Degree $2$
Conductor $65$
Sign $-0.444 - 0.895i$
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  + (−1.09 + 1.89i)2-s + (0.866 + 0.5i)3-s + (−1.39 − 2.41i)4-s + (0.456 + 2.18i)5-s + (−1.89 + 1.09i)6-s + (−0.866 − 1.5i)7-s + 1.73·8-s + (−1 − 1.73i)9-s + (−4.64 − 1.52i)10-s + (2.29 + 1.32i)11-s − 2.79i·12-s + (3.46 + i)13-s + 3.79·14-s + (−0.698 + 2.12i)15-s + (0.895 − 1.55i)16-s + (3.96 − 2.29i)17-s + ⋯
L(s)  = 1  + (−0.773 + 1.34i)2-s + (0.499 + 0.288i)3-s + (−0.697 − 1.20i)4-s + (0.204 + 0.978i)5-s + (−0.773 + 0.446i)6-s + (−0.327 − 0.566i)7-s + 0.612·8-s + (−0.333 − 0.577i)9-s + (−1.47 − 0.483i)10-s + (0.690 + 0.398i)11-s − 0.805i·12-s + (0.960 + 0.277i)13-s + 1.01·14-s + (−0.180 + 0.548i)15-s + (0.223 − 0.387i)16-s + (0.962 − 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.444 - 0.895i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.374293 + 0.603929i\)
\(L(\frac12)\) \(\approx\) \(0.374293 + 0.603929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.456 - 2.18i)T \)
13 \( 1 + (-3.46 - i)T \)
good2 \( 1 + (1.09 - 1.89i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.866 + 1.5i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.29 - 1.32i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.96 + 2.29i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.96 + 2.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.29 - 3.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.66iT - 31T^{2} \)
37 \( 1 + (3.96 - 6.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.29 + 1.32i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.22 - 0.708i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.75T + 47T^{2} \)
53 \( 1 - 1.58iT - 53T^{2} \)
59 \( 1 + (2.91 - 1.68i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.29 - 9.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.43 + 12.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.08 - 1.77i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + (-3.70 - 2.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.23 - 3.87i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.19391143268521624322929659817, −14.55341307242264965730574595052, −13.74604611369796729056651685331, −11.76201311025281487204335182812, −10.14826851177056650503723001709, −9.374039989802255794666728885414, −8.137045358244951765923741190796, −6.88972277598566996675735052984, −6.07842581584673251710227019933, −3.59127200802506070753869368518, 1.69638152110125240897701556347, 3.48460614171024375361892633897, 5.77957670439194024130564255707, 8.253820803741740139096562458366, 8.779270311953487497167977757023, 9.890313229600617665493659581269, 11.17586717901964810910794710250, 12.25759715205384997351520053023, 13.05388130151919462044751811049, 14.17316639557618164860823231296

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