Properties

Label 2-65-5.4-c1-0-4
Degree $2$
Conductor $65$
Sign $0.139 + 0.990i$
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  − 1.21i·2-s − 1.31i·3-s + 0.525·4-s + (−2.21 + 0.311i)5-s − 1.59·6-s + 2.90i·7-s − 3.06i·8-s + 1.28·9-s + (0.377 + 2.68i)10-s + 0.214·11-s − 0.688i·12-s + i·13-s + 3.52·14-s + (0.407 + 2.90i)15-s − 2.67·16-s + 6.42i·17-s + ⋯
L(s)  = 1  − 0.858i·2-s − 0.756i·3-s + 0.262·4-s + (−0.990 + 0.139i)5-s − 0.649·6-s + 1.09i·7-s − 1.08i·8-s + 0.426·9-s + (0.119 + 0.850i)10-s + 0.0646·11-s − 0.198i·12-s + 0.277i·13-s + 0.942·14-s + (0.105 + 0.749i)15-s − 0.668·16-s + 1.55i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.139 + 0.990i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.139 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682182 - 0.593037i\)
\(L(\frac12)\) \(\approx\) \(0.682182 - 0.593037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.21 - 0.311i)T \)
13 \( 1 - iT \)
good2 \( 1 + 1.21iT - 2T^{2} \)
3 \( 1 + 1.31iT - 3T^{2} \)
7 \( 1 - 2.90iT - 7T^{2} \)
11 \( 1 - 0.214T + 11T^{2} \)
17 \( 1 - 6.42iT - 17T^{2} \)
19 \( 1 + 2.21T + 19T^{2} \)
23 \( 1 + 4.68iT - 23T^{2} \)
29 \( 1 + 8.70T + 29T^{2} \)
31 \( 1 + 5.59T + 31T^{2} \)
37 \( 1 - 2.28iT - 37T^{2} \)
41 \( 1 - 3.05T + 41T^{2} \)
43 \( 1 + 6.36iT - 43T^{2} \)
47 \( 1 - 1.09iT - 47T^{2} \)
53 \( 1 - 6.23iT - 53T^{2} \)
59 \( 1 - 9.26T + 59T^{2} \)
61 \( 1 + 0.280T + 61T^{2} \)
67 \( 1 + 7.76iT - 67T^{2} \)
71 \( 1 + 6.08T + 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 9.52iT - 83T^{2} \)
89 \( 1 - 5.61T + 89T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−14.83732895583348212073964348577, −12.84374604992164036909273472563, −12.47028727162652389162424390978, −11.49078673923172472737254844609, −10.47341383694129709071912947934, −8.812783821490400981211744299532, −7.47957343437731933098144318417, −6.27684157815518644741708600953, −3.90029575512682996202499098165, −2.07460690454259973849915542594, 3.78286193813014265008327954860, 5.12333742502535471730544271856, 7.09411328894992678105953640426, 7.67243354238388677741909016302, 9.311827547440177808522265361757, 10.75183512733246721045744708585, 11.53510771773781257337920270730, 13.14180825694376987059746699447, 14.53285538547678360809346543414, 15.32903887335426770783810270537

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