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Properties

Label 2-364-364.55-c1-0-12
Degree $2$
Conductor $364$
Sign $-0.651 - 0.758i$
Analytic cond. $2.90655$
Root an. cond. $1.70486$
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  + (−0.895 + 1.09i)2-s + (1.32 + 2.29i)3-s + (−0.395 − 1.96i)4-s − i·5-s + (−3.69 − 0.604i)6-s + (2.29 + 1.32i)7-s + (2.49 + 1.32i)8-s + (−2 + 3.46i)9-s + (1.09 + 0.895i)10-s + (3.96 − 3.49i)12-s + (−2.59 + 2.5i)13-s + (−3.49 + 1.32i)14-s + (2.29 − 1.32i)15-s + (−3.68 + 1.55i)16-s + (3.46 + 2i)17-s + (−1.99 − 5.29i)18-s + ⋯
L(s)  = 1  + (−0.633 + 0.773i)2-s + (0.763 + 1.32i)3-s + (−0.197 − 0.980i)4-s − 0.447i·5-s + (−1.50 − 0.246i)6-s + (0.866 + 0.499i)7-s + (0.883 + 0.467i)8-s + (−0.666 + 1.15i)9-s + (0.346 + 0.283i)10-s + (1.14 − 1.01i)12-s + (−0.720 + 0.693i)13-s + (−0.935 + 0.353i)14-s + (0.591 − 0.341i)15-s + (−0.921 + 0.387i)16-s + (0.840 + 0.485i)17-s + (−0.471 − 1.24i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(364\)    =    \(2^{2} \cdot 7 \cdot 13\)
Sign: $-0.651 - 0.758i$
Analytic conductor: \(2.90655\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{364} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 364,\ (\ :1/2),\ -0.651 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.535753 + 1.16681i\)
\(L(\frac12)\) \(\approx\) \(0.535753 + 1.16681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.895 - 1.09i)T \)
7 \( 1 + (-2.29 - 1.32i)T \)
13 \( 1 + (2.59 - 2.5i)T \)
good3 \( 1 + (-1.32 - 2.29i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + iT - 5T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.64 - 4.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.29 + 1.32i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4 + 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.73 + i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.58 + 2.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-3.96 + 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.58 + 2.64i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.29 + 1.32i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 5.29iT - 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + (-12.1 + 7i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.1 + 7i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.42180610048890283325792608418, −10.41490456810268182670816466536, −9.734432401310432928587534122716, −8.849779599998073160209682922103, −8.387706154426143758811677830004, −7.36437196545003431162610087590, −5.75475778695229415010168222526, −4.90517566859179913132916526640, −3.98141717784742200464360186786, −2.03183250527285891168741080489, 1.09820828921063432562835813670, 2.37201366860471064872016034342, 3.32494784240088333673747369785, 5.01383766193584606159292121362, 7.13195149262260718434757404505, 7.26482277096407286438863798177, 8.291560863073082427760567637642, 9.052770554970227153527011323210, 10.30071346821393827917605279210, 11.08970099581568772037501425188

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