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Properties

Label 2-840-21.17-c1-0-28
Degree $2$
Conductor $840$
Sign $-0.273 + 0.961i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
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  + (1.46 − 0.927i)3-s + (0.5 + 0.866i)5-s + (−1.49 − 2.18i)7-s + (1.27 − 2.71i)9-s + (−5.52 − 3.19i)11-s + 0.186i·13-s + (1.53 + 0.803i)15-s + (3.85 − 6.67i)17-s + (−5.08 + 2.93i)19-s + (−4.21 − 1.79i)21-s + (0.256 − 0.148i)23-s + (−0.499 + 0.866i)25-s + (−0.644 − 5.15i)27-s − 8.18i·29-s + (5.76 + 3.33i)31-s + ⋯
L(s)  = 1  + (0.844 − 0.535i)3-s + (0.223 + 0.387i)5-s + (−0.566 − 0.824i)7-s + (0.426 − 0.904i)9-s + (−1.66 − 0.962i)11-s + 0.0516i·13-s + (0.396 + 0.207i)15-s + (0.934 − 1.61i)17-s + (−1.16 + 0.673i)19-s + (−0.919 − 0.392i)21-s + (0.0535 − 0.0308i)23-s + (−0.0999 + 0.173i)25-s + (−0.123 − 0.992i)27-s − 1.51i·29-s + (1.03 + 0.598i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.273 + 0.961i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.273 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.981229 - 1.29867i\)
\(L(\frac12)\) \(\approx\) \(0.981229 - 1.29867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.46 + 0.927i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.49 + 2.18i)T \)
good11 \( 1 + (5.52 + 3.19i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.186iT - 13T^{2} \)
17 \( 1 + (-3.85 + 6.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.08 - 2.93i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.256 + 0.148i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.18iT - 29T^{2} \)
31 \( 1 + (-5.76 - 3.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.69 - 2.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.67T + 41T^{2} \)
43 \( 1 + 0.268T + 43T^{2} \)
47 \( 1 + (-2.93 - 5.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.72 - 1.57i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.55 - 6.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.46 + 3.15i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.460 + 0.798i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.44iT - 71T^{2} \)
73 \( 1 + (5.90 + 3.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.455 - 0.788i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.79T + 83T^{2} \)
89 \( 1 + (3.80 + 6.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−10.03525022992653820016905473525, −9.120424450297065368797932063401, −7.923364865137067800514825590708, −7.68268949389873376337819759684, −6.57438326682003484558207382922, −5.76743404255842916794010769937, −4.35174631704222925540125527324, −3.11369967816581131152838310460, −2.56702415901119464342390591055, −0.69060713185827487948930331007, 2.06550561605774927410241859025, 2.83663144096284549478871222538, 4.10605436939603745860983643457, 5.08923880017341211409444930076, 5.89235272992939802069174275441, 7.20719868475162840441060642413, 8.197191721056943729444938238537, 8.674238921402480195411416551346, 9.679145722555021964027645936728, 10.23851190785090059906488749992

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