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Properties

Label 2-840-168.5-c1-0-93
Degree $2$
Conductor $840$
Sign $-0.0525 + 0.998i$
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  + (−1.27 − 0.609i)2-s + (1.29 − 1.14i)3-s + (1.25 + 1.55i)4-s + (0.866 + 0.5i)5-s + (−2.35 + 0.677i)6-s + (2.04 − 1.67i)7-s + (−0.655 − 2.75i)8-s + (0.356 − 2.97i)9-s + (−0.800 − 1.16i)10-s + (−0.791 − 1.37i)11-s + (3.41 + 0.569i)12-s + 2.55·13-s + (−3.63 + 0.887i)14-s + (1.69 − 0.348i)15-s + (−0.840 + 3.91i)16-s + (0.000650 + 0.00112i)17-s + ⋯
L(s)  = 1  + (−0.902 − 0.431i)2-s + (0.747 − 0.663i)3-s + (0.628 + 0.777i)4-s + (0.387 + 0.223i)5-s + (−0.960 + 0.276i)6-s + (0.774 − 0.632i)7-s + (−0.231 − 0.972i)8-s + (0.118 − 0.992i)9-s + (−0.253 − 0.368i)10-s + (−0.238 − 0.413i)11-s + (0.986 + 0.164i)12-s + 0.708·13-s + (−0.971 + 0.237i)14-s + (0.438 − 0.0898i)15-s + (−0.210 + 0.977i)16-s + (0.000157 + 0.000273i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0525 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0525 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0525 + 0.998i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.0525 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05901 - 1.11621i\)
\(L(\frac12)\) \(\approx\) \(1.05901 - 1.11621i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 + 0.609i)T \)
3 \( 1 + (-1.29 + 1.14i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-2.04 + 1.67i)T \)
good11 \( 1 + (0.791 + 1.37i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.55T + 13T^{2} \)
17 \( 1 + (-0.000650 - 0.00112i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.482 + 0.836i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.79 + 1.61i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.84T + 29T^{2} \)
31 \( 1 + (2.05 - 1.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.97 - 5.18i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.33T + 41T^{2} \)
43 \( 1 - 5.61iT - 43T^{2} \)
47 \( 1 + (-5.79 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.58 - 7.94i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.29 - 1.90i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.28 + 7.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.97 - 4.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.45iT - 71T^{2} \)
73 \( 1 + (7.92 - 4.57i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.16 + 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.94iT - 83T^{2} \)
89 \( 1 + (7.80 - 13.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.1iT - 97T^{2} \)
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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

−9.931374915308895182904044695990, −9.032791034590013027885086935343, −8.294173193683081795533283479783, −7.68264210995954188812411674127, −6.86069521153195449721951763294, −5.93151160857826711599805498367, −4.17997577593499653636556174593, −3.14690784282940633290938979647, −2.05637472355215631625060431667, −1.00321850494855195275876281488, 1.66117662961816078611424061681, 2.57594658821291352850589548016, 4.16788577333358170502095517198, 5.31299767028479673201275719002, 5.95328151700985587816741677093, 7.40407531540500010460234958293, 8.033666298706888256871836838327, 8.836667216986144389547544022486, 9.361779605247187522535826276562, 10.18161499156765732287160163980

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