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Properties

Label 2-65-65.58-c1-0-4
Degree $2$
Conductor $65$
Sign $0.716 + 0.697i$
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  + (−0.885 − 0.511i)2-s + (2.69 − 0.721i)3-s + (−0.477 − 0.826i)4-s + (−1.45 + 1.69i)5-s + (−2.75 − 0.737i)6-s + (−0.481 − 0.834i)7-s + 3.02i·8-s + (4.12 − 2.38i)9-s + (2.15 − 0.756i)10-s + (1.60 − 0.430i)11-s + (−1.88 − 1.88i)12-s + (−3.11 + 1.82i)13-s + 0.985i·14-s + (−2.70 + 5.61i)15-s + (0.590 − 1.02i)16-s + (−1.87 + 7.00i)17-s + ⋯
L(s)  = 1  + (−0.626 − 0.361i)2-s + (1.55 − 0.416i)3-s + (−0.238 − 0.413i)4-s + (−0.651 + 0.758i)5-s + (−1.12 − 0.301i)6-s + (−0.182 − 0.315i)7-s + 1.06i·8-s + (1.37 − 0.794i)9-s + (0.682 − 0.239i)10-s + (0.484 − 0.129i)11-s + (−0.542 − 0.542i)12-s + (−0.863 + 0.505i)13-s + 0.263i·14-s + (−0.697 + 1.45i)15-s + (0.147 − 0.255i)16-s + (−0.455 + 1.69i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.805890 - 0.327780i\)
\(L(\frac12)\) \(\approx\) \(0.805890 - 0.327780i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.45 - 1.69i)T \)
13 \( 1 + (3.11 - 1.82i)T \)
good2 \( 1 + (0.885 + 0.511i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-2.69 + 0.721i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.481 + 0.834i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.60 + 0.430i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.87 - 7.00i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.707 + 2.64i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.997 + 3.72i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.253 + 0.146i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.125 - 0.125i)T - 31iT^{2} \)
37 \( 1 + (-2.04 + 3.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.79 + 6.69i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.67 - 2.05i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 7.84T + 47T^{2} \)
53 \( 1 + (1.99 + 1.99i)T + 53iT^{2} \)
59 \( 1 + (4.87 + 1.30i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.04 + 1.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.32 - 3.64i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.6 - 3.37i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 3.22iT - 73T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 + 8.56T + 83T^{2} \)
89 \( 1 + (0.134 + 0.500i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.50 - 3.75i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.55036504273756431865376752617, −14.05978959208263581455785807054, −12.67660287916855698000343531976, −11.13454208299848522117141659502, −10.00192865265243891124744714445, −8.906926674853204203129669672299, −7.995468627506426307117506347781, −6.73782687734717784618244104048, −3.99280070110371800721017247755, −2.28315081403106508849276899203, 3.18862487127563121515299737438, 4.56690591824306209546167592923, 7.39185193894824845891714204690, 8.117007724052636214333010385313, 9.234387446448613711034229191962, 9.629747786509097590437688531249, 11.91076830060501431575489792913, 12.98581666408136900813385250666, 14.04811702492785080378791069601, 15.30433925196835711748019193113

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