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Properties

Label 2-65-65.49-c1-0-0
Degree $2$
Conductor $65$
Sign $0.0496 - 0.998i$
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 + (3.64 − 3.26i)10-s + (2.29 − 1.32i)11-s − 2.79i·12-s + (−3.46 + i)13-s + 3.79·14-s + (1.49 + 1.66i)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 + (1.15 − 1.03i)10-s + (0.690 − 0.398i)11-s − 0.805i·12-s + (−0.960 + 0.277i)13-s + 1.01·14-s + (0.384 + 0.430i)15-s + (0.223 + 0.387i)16-s + (−0.962 − 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0496 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0496 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.0496 - 0.998i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.0496 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782517 + 0.744614i\)
\(L(\frac12)\) \(\approx\) \(0.782517 + 0.744614i\)
\(L(\frac{3}{2})\) 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

−15.28123783962322113001102024792, −14.11724877003752881374292648389, −13.36844856288419686246365267428, −12.08407516122359812401698076863, −10.84865745176667388353848921192, −9.014045517182116526895184205477, −7.82059247729527947090012574869, −6.55411531417451381422625263340, −5.06021038345619471715466873224, −4.45173881883099361984777950509, 2.38666506630074288429616164973, 3.99893193282605464660449145949, 5.71215355939499884915873279929, 7.18865746680236474386755340585, 9.283672684461592407447255030458, 10.64801815438318370370499127532, 11.50930154453640985550139575977, 12.15488956004675518804324702256, 13.18209064809670799373048486308, 14.70026421731602382944326940953

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