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Properties

Label 2-65-13.4-c1-0-2
Degree $2$
Conductor $65$
Sign $-0.593 + 0.804i$
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  + (−2.16 − 1.24i)2-s + (1.41 − 2.44i)3-s + (2.11 + 3.66i)4-s − i·5-s + (−6.10 + 3.52i)6-s + (−1.64 + 0.952i)7-s − 5.55i·8-s + (−2.49 − 4.32i)9-s + (−1.24 + 2.16i)10-s + (0.926 + 0.534i)11-s + 11.9·12-s + (1.40 + 3.32i)13-s + 4.75·14-s + (−2.44 − 1.41i)15-s + (−2.70 + 4.69i)16-s + (0.318 + 0.551i)17-s + ⋯
L(s)  = 1  + (−1.52 − 0.882i)2-s + (0.816 − 1.41i)3-s + (1.05 + 1.83i)4-s − 0.447i·5-s + (−2.49 + 1.43i)6-s + (−0.623 + 0.360i)7-s − 1.96i·8-s + (−0.831 − 1.44i)9-s + (−0.394 + 0.683i)10-s + (0.279 + 0.161i)11-s + 3.44·12-s + (0.388 + 0.921i)13-s + 1.27·14-s + (−0.632 − 0.364i)15-s + (−0.677 + 1.17i)16-s + (0.0772 + 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.593 + 0.804i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.593 + 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253878 - 0.502645i\)
\(L(\frac12)\) \(\approx\) \(0.253878 - 0.502645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
13 \( 1 + (-1.40 - 3.32i)T \)
good2 \( 1 + (2.16 + 1.24i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.41 + 2.44i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.64 - 0.952i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.926 - 0.534i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.318 - 0.551i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.96 + 2.86i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.90 + 3.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.72 - 8.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 + (-0.655 - 0.378i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.232 + 0.133i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.318 - 0.551i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.44iT - 47T^{2} \)
53 \( 1 + 6.99T + 53T^{2} \)
59 \( 1 + (0.641 - 0.370i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.09 + 3.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.01 + 4.04i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.45 + 4.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.71iT - 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 - 5.11iT - 83T^{2} \)
89 \( 1 + (10.8 + 6.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.65 - 2.11i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.22329102953425615021118316741, −12.94831004157138800030883729464, −12.28053742195412289460588610392, −11.20221260677500624781644267718, −9.387034785891927352362762674841, −8.926078177384048536639686941504, −7.73796348035818052618654332204, −6.70790455546909363989073532389, −2.99578752465532891473641690173, −1.47694879281105926563825759706, 3.44960781111466495319585531492, 5.74789291303059430250809435651, 7.39259802992141814342508989946, 8.467928918367636435124085718426, 9.656169692112918419661671502619, 10.03414202180356788996274922473, 11.15735950202928408425135072040, 13.67150442853710266738381457444, 14.83641727829813048398514160042, 15.53341881481221786280921006039

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