Properties

Label 2-280-35.13-c1-0-0
Degree $2$
Conductor $280$
Sign $-0.648 - 0.761i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (−0.0703 + 0.0703i)3-s + (−2.16 + 0.574i)5-s + (−2.58 + 0.562i)7-s + 2.99i·9-s + 0.777·11-s + (−3.93 + 3.93i)13-s + (0.111 − 0.192i)15-s + (0.982 + 0.982i)17-s − 1.14·19-s + (0.142 − 0.221i)21-s + (−1.46 − 1.46i)23-s + (4.34 − 2.48i)25-s + (−0.421 − 0.421i)27-s + 4.69i·29-s − 6.45i·31-s + ⋯
L(s)  = 1  + (−0.0405 + 0.0405i)3-s + (−0.966 + 0.256i)5-s + (−0.977 + 0.212i)7-s + 0.996i·9-s + 0.234·11-s + (−1.09 + 1.09i)13-s + (0.0288 − 0.0496i)15-s + (0.238 + 0.238i)17-s − 0.263·19-s + (0.0310 − 0.0482i)21-s + (−0.304 − 0.304i)23-s + (0.868 − 0.496i)25-s + (−0.0810 − 0.0810i)27-s + 0.870i·29-s − 1.15i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.648 - 0.761i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.648 - 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.239629 + 0.519140i\)
\(L(\frac12)\) \(\approx\) \(0.239629 + 0.519140i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.16 - 0.574i)T \)
7 \( 1 + (2.58 - 0.562i)T \)
good3 \( 1 + (0.0703 - 0.0703i)T - 3iT^{2} \)
11 \( 1 - 0.777T + 11T^{2} \)
13 \( 1 + (3.93 - 3.93i)T - 13iT^{2} \)
17 \( 1 + (-0.982 - 0.982i)T + 17iT^{2} \)
19 \( 1 + 1.14T + 19T^{2} \)
23 \( 1 + (1.46 + 1.46i)T + 23iT^{2} \)
29 \( 1 - 4.69iT - 29T^{2} \)
31 \( 1 + 6.45iT - 31T^{2} \)
37 \( 1 + (1.30 - 1.30i)T - 37iT^{2} \)
41 \( 1 - 9.81iT - 41T^{2} \)
43 \( 1 + (7.13 + 7.13i)T + 43iT^{2} \)
47 \( 1 + (-7.34 - 7.34i)T + 47iT^{2} \)
53 \( 1 + (2.08 + 2.08i)T + 53iT^{2} \)
59 \( 1 - 8.29T + 59T^{2} \)
61 \( 1 + 5.88iT - 61T^{2} \)
67 \( 1 + (6.30 - 6.30i)T - 67iT^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (-7.23 + 7.23i)T - 73iT^{2} \)
79 \( 1 - 2.83iT - 79T^{2} \)
83 \( 1 + (10.4 - 10.4i)T - 83iT^{2} \)
89 \( 1 - 3.41T + 89T^{2} \)
97 \( 1 + (-8.50 - 8.50i)T + 97iT^{2} \)
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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

−12.14022125076558984591761276869, −11.35241440194788019603514880922, −10.33100690217455832114288118289, −9.425349658933037703967395062768, −8.285644756452227228379599245191, −7.29958430265785302346793932854, −6.45083218492130518716973199758, −4.94643648358127694125523093418, −3.85404939569987442786058649226, −2.45287489582426247986309839198, 0.42009032164349763740034057266, 3.06624223557928701608013514503, 4.00258851473475914398238046382, 5.42635424488939854036002922907, 6.72229991318820788455258472182, 7.51853949681672417058626194344, 8.666440384544602890468987793693, 9.659022797953064780877590810349, 10.47462456245219395991875371644, 11.83357851512005536640844407317

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