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Properties

Label 2-280-280.107-c1-0-33
Degree $2$
Conductor $280$
Sign $-0.224 + 0.974i$
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  + (−0.626 + 1.26i)2-s + (0.182 + 0.0488i)3-s + (−1.21 − 1.58i)4-s + (−1.60 + 1.55i)5-s + (−0.176 + 0.200i)6-s + (−1.91 − 1.82i)7-s + (2.77 − 0.543i)8-s + (−2.56 − 1.48i)9-s + (−0.968 − 3.01i)10-s + (1.05 + 1.83i)11-s + (−0.143 − 0.349i)12-s + (−3.49 − 3.49i)13-s + (3.51 − 1.28i)14-s + (−0.368 + 0.205i)15-s + (−1.04 + 3.85i)16-s + (0.955 − 3.56i)17-s + ⋯
L(s)  = 1  + (−0.443 + 0.896i)2-s + (0.105 + 0.0282i)3-s + (−0.607 − 0.794i)4-s + (−0.717 + 0.696i)5-s + (−0.0719 + 0.0818i)6-s + (−0.724 − 0.689i)7-s + (0.981 − 0.192i)8-s + (−0.855 − 0.494i)9-s + (−0.306 − 0.951i)10-s + (0.318 + 0.551i)11-s + (−0.0415 − 0.100i)12-s + (−0.969 − 0.969i)13-s + (0.938 − 0.344i)14-s + (−0.0951 + 0.0530i)15-s + (−0.262 + 0.964i)16-s + (0.231 − 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.224 + 0.974i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.103411 - 0.129946i\)
\(L(\frac12)\) \(\approx\) \(0.103411 - 0.129946i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.626 - 1.26i)T \)
5 \( 1 + (1.60 - 1.55i)T \)
7 \( 1 + (1.91 + 1.82i)T \)
good3 \( 1 + (-0.182 - 0.0488i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.05 - 1.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.49 + 3.49i)T + 13iT^{2} \)
17 \( 1 + (-0.955 + 3.56i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.681 + 0.393i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.51 - 2.28i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + (3.12 - 1.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.885 - 3.30i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.47T + 41T^{2} \)
43 \( 1 + (1.31 - 1.31i)T - 43iT^{2} \)
47 \( 1 + (1.16 + 4.34i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.376 + 1.40i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.46 - 3.15i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.644 + 0.372i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.18 - 11.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.25iT - 71T^{2} \)
73 \( 1 + (10.9 + 2.92i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.89 + 6.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.14 - 8.14i)T - 83iT^{2} \)
89 \( 1 + (4.78 + 2.76i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.2 + 12.2i)T + 97iT^{2} \)
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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

−11.55251993395300443030440883161, −10.19019058284794668399321905534, −9.853422330710986649198458508075, −8.506138640556632155871953535680, −7.54652237303193680332427044491, −6.88245750213228836944967412037, −5.83085273204029276743329770642, −4.37703739697667693371560804009, −3.06639528149473573119989679367, −0.13649047278901882920964141649, 2.16833800249038587307452463030, 3.52614045635028503015126980207, 4.67340944925292711252850395537, 6.09695132897228750532689684123, 7.73462342372461546776821381016, 8.540523553676800976754293491809, 9.188845357555383258444837855712, 10.22413249989013000259530555376, 11.38724924445820678677094584388, 12.10192563600002937217947870041

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