Properties

Label 2-65-13.10-c1-0-3
Degree $2$
Conductor $65$
Sign $0.484 + 0.875i$
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  + (1.05 − 0.609i)2-s + (−1.16 − 2.01i)3-s + (−0.256 + 0.443i)4-s i·5-s + (−2.46 − 1.42i)6-s + (3.11 + 1.80i)7-s + 3.06i·8-s + (−1.21 + 2.11i)9-s + (−0.609 − 1.05i)10-s + (−4.65 + 2.68i)11-s + 1.19·12-s + (1.81 − 3.11i)13-s + 4.39·14-s + (−2.01 + 1.16i)15-s + (1.35 + 2.34i)16-s + (−0.565 + 0.980i)17-s + ⋯
L(s)  = 1  + (0.746 − 0.431i)2-s + (−0.673 − 1.16i)3-s + (−0.128 + 0.221i)4-s − 0.447i·5-s + (−1.00 − 0.580i)6-s + (1.17 + 0.680i)7-s + 1.08i·8-s + (−0.406 + 0.704i)9-s + (−0.192 − 0.334i)10-s + (−1.40 + 0.809i)11-s + 0.344·12-s + (0.504 − 0.863i)13-s + 1.17·14-s + (−0.521 + 0.301i)15-s + (0.339 + 0.587i)16-s + (−0.137 + 0.237i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.484 + 0.875i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.484 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895026 - 0.527727i\)
\(L(\frac12)\) \(\approx\) \(0.895026 - 0.527727i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 3.11i)T \)
good2 \( 1 + (-1.05 + 0.609i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.16 + 2.01i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3.11 - 1.80i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.65 - 2.68i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.565 - 0.980i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.94 + 3.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0123 - 0.0214i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
37 \( 1 + (-7.53 + 4.35i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.23 + 1.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.565 - 0.980i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.58iT - 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 + (0.148 + 0.0857i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.68 - 2.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.54 + 3.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.35 - 5.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.70iT - 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 + (-13.9 + 8.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.5 + 6.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.42302680234317028274053547927, −12.97207392389289149231656221357, −12.76740594947765634018058348632, −11.76531640754269803092495395836, −10.75958084177328027005701448713, −8.456886944885747281753485335411, −7.68422755443374245791796019238, −5.73154363345460667563288823068, −4.76562561577916403878006924316, −2.21990749613258928673505807673, 4.03098402667033651751282679777, 4.99175200570362950331213995770, 6.08625732330117526726608202729, 7.84857580399543244253049700742, 9.704895302553094668753371638454, 10.77143993004578887043044939915, 11.33012638308623222711387126149, 13.33103850557207692695489901720, 14.12525767538565813791760632012, 15.13810026655952637233270399102

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