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Properties

Label 2-65-65.28-c1-0-4
Degree $2$
Conductor $65$
Sign $0.792 + 0.609i$
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  + (1.31 − 0.759i)2-s + (0.175 − 0.653i)3-s + (0.152 − 0.263i)4-s + (−2.15 − 0.600i)5-s + (−0.265 − 0.991i)6-s + (−1.29 + 2.24i)7-s + 2.57i·8-s + (2.20 + 1.27i)9-s + (−3.28 + 0.845i)10-s + (1.29 − 4.82i)11-s + (−0.145 − 0.145i)12-s + (−2.37 − 2.71i)13-s + 3.93i·14-s + (−0.769 + 1.30i)15-s + (2.25 + 3.91i)16-s + (0.0790 − 0.0211i)17-s + ⋯
L(s)  = 1  + (0.929 − 0.536i)2-s + (0.101 − 0.377i)3-s + (0.0761 − 0.131i)4-s + (−0.963 − 0.268i)5-s + (−0.108 − 0.404i)6-s + (−0.490 + 0.849i)7-s + 0.910i·8-s + (0.733 + 0.423i)9-s + (−1.03 + 0.267i)10-s + (0.390 − 1.45i)11-s + (−0.0420 − 0.0420i)12-s + (−0.658 − 0.752i)13-s + 1.05i·14-s + (−0.198 + 0.336i)15-s + (0.564 + 0.977i)16-s + (0.0191 − 0.00513i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.792 + 0.609i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.792 + 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13022 - 0.384216i\)
\(L(\frac12)\) \(\approx\) \(1.13022 - 0.384216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.15 + 0.600i)T \)
13 \( 1 + (2.37 + 2.71i)T \)
good2 \( 1 + (-1.31 + 0.759i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.175 + 0.653i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.29 - 2.24i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.29 + 4.82i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.0790 + 0.0211i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.71 - 0.726i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.91 + 1.05i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.31 + 2.49i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.32 - 2.32i)T - 31iT^{2} \)
37 \( 1 + (0.285 + 0.494i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.0 - 2.69i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.0354 - 0.132i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 2.30T + 47T^{2} \)
53 \( 1 + (-6.70 - 6.70i)T + 53iT^{2} \)
59 \( 1 + (0.694 + 2.59i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.74 - 4.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.6 - 7.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.98 + 7.42i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 6.61iT - 73T^{2} \)
79 \( 1 + 5.71iT - 79T^{2} \)
83 \( 1 - 3.70T + 83T^{2} \)
89 \( 1 + (17.2 + 4.63i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.65 - 2.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.57495514542939837289480593103, −13.39983061301076602649126441707, −12.51577756443096479804271676125, −11.92392811090760477542103179422, −10.67505587459458737370221831995, −8.754066786401803656834500774982, −7.79476726228576268921747297896, −5.89476724545944050483016158509, −4.34366041974543086089987128647, −2.90650553003474953498185446163, 3.95745940483377041925926730570, 4.49956729883076661572392720432, 6.72524566811793457796565193959, 7.30380421153116645098024302015, 9.490138731491951046442117701158, 10.34763599730206012885872099020, 12.09380489874650016083662645168, 12.85578651659278866718492321747, 14.25497826894405155403355670461, 14.95879517144155210948356296317

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