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Properties

Label 2-280-280.213-c1-0-40
Degree $2$
Conductor $280$
Sign $-0.683 - 0.730i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−0.5 + 0.133i)3-s + 2i·4-s + (−0.133 − 2.23i)5-s + (0.633 + 0.366i)6-s + (−2.5 + 0.866i)7-s + (2 − 2i)8-s + (−2.36 + 1.36i)9-s + (−2.09 + 2.36i)10-s + (2.36 + 1.36i)11-s + (−0.267 − i)12-s + (−3 − 3i)13-s + (3.36 + 1.63i)14-s + (0.366 + 1.09i)15-s − 4·16-s + (−4.09 + 1.09i)17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.288 + 0.0773i)3-s + i·4-s + (−0.0599 − 0.998i)5-s + (0.258 + 0.149i)6-s + (−0.944 + 0.327i)7-s + (0.707 − 0.707i)8-s + (−0.788 + 0.455i)9-s + (−0.663 + 0.748i)10-s + (0.713 + 0.411i)11-s + (−0.0773 − 0.288i)12-s + (−0.832 − 0.832i)13-s + (0.899 + 0.436i)14-s + (0.0945 + 0.283i)15-s − 16-s + (−0.993 + 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.683 - 0.730i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.683 - 0.730i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 + (0.133 + 2.23i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
good3 \( 1 + (0.5 - 0.133i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-2.36 - 1.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (4.09 - 1.09i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 5.59i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 + (0.169 + 0.0980i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.464 + 1.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.46iT - 41T^{2} \)
43 \( 1 + (0.633 - 0.633i)T - 43iT^{2} \)
47 \( 1 + (-2.36 + 8.83i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3 + 11.1i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.26 + 2.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.96 + 3.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.96 - 2.13i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.53T + 71T^{2} \)
73 \( 1 + (2.90 + 10.8i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.803 - 0.464i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.56 - 7.56i)T + 83iT^{2} \)
89 \( 1 + (3.40 + 5.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.73 - 9.73i)T + 97iT^{2} \)
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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

−11.35047523241033052477368768580, −10.18581632910974073149982339875, −9.411702831550052466275725029017, −8.638380396110058048965093890751, −7.67743553580030210321353504033, −6.27963831145766162924715230676, −4.97854142869800318530993248435, −3.64600100804541746265500342734, −2.11489822527511908787053687211, 0, 2.59475920710829249919940461625, 4.29332914125950315377178814389, 6.08168498421144830718976732131, 6.55945260221215045398144786333, 7.31159751016880320366546950420, 8.809742612845560140779735292210, 9.427123086148708300677871491120, 10.55522934305167770215341721720, 11.21490028457205264367185592494

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