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Properties

Label 2-840-35.17-c1-0-19
Degree $2$
Conductor $840$
Sign $0.530 + 0.847i$
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  + (0.965 − 0.258i)3-s + (2.13 − 0.652i)5-s + (−2.13 + 1.55i)7-s + (0.866 − 0.499i)9-s + (2.71 − 4.69i)11-s + (−4.87 − 4.87i)13-s + (1.89 − 1.18i)15-s + (−1.79 − 6.70i)17-s + (1.02 + 1.77i)19-s + (−1.66 + 2.05i)21-s + (5.67 + 1.52i)23-s + (4.14 − 2.79i)25-s + (0.707 − 0.707i)27-s + 5.88i·29-s + (7.01 + 4.04i)31-s + ⋯
L(s)  = 1  + (0.557 − 0.149i)3-s + (0.956 − 0.292i)5-s + (−0.808 + 0.588i)7-s + (0.288 − 0.166i)9-s + (0.817 − 1.41i)11-s + (−1.35 − 1.35i)13-s + (0.489 − 0.305i)15-s + (−0.435 − 1.62i)17-s + (0.234 + 0.406i)19-s + (−0.362 + 0.449i)21-s + (1.18 + 0.317i)23-s + (0.829 − 0.558i)25-s + (0.136 − 0.136i)27-s + 1.09i·29-s + (1.25 + 0.727i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73871 - 0.963605i\)
\(L(\frac12)\) \(\approx\) \(1.73871 - 0.963605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-2.13 + 0.652i)T \)
7 \( 1 + (2.13 - 1.55i)T \)
good11 \( 1 + (-2.71 + 4.69i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.87 + 4.87i)T + 13iT^{2} \)
17 \( 1 + (1.79 + 6.70i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.02 - 1.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.67 - 1.52i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.88iT - 29T^{2} \)
31 \( 1 + (-7.01 - 4.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.108 - 0.406i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.34iT - 41T^{2} \)
43 \( 1 + (-3.69 + 3.69i)T - 43iT^{2} \)
47 \( 1 + (-0.806 - 0.216i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.502 - 1.87i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.69 - 8.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.93 - 4.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.66 - 2.58i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.152T + 71T^{2} \)
73 \( 1 + (-4.37 + 1.17i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.88 + 2.82i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.533 + 0.533i)T + 83iT^{2} \)
89 \( 1 + (-0.318 - 0.552i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.20 - 2.20i)T - 97iT^{2} \)
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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

−9.854728279948417893269608006176, −9.127828744831070399365287147347, −8.725298802990515765725070188007, −7.45770892921789484925968504584, −6.58513553895861276521060871092, −5.65350006438257690215424168696, −4.91037453027833284592210784832, −3.06085573545124881866955789885, −2.79368375668675812151697370225, −0.953927149488673951381708777216, 1.76578788504587043995033766783, 2.64541603956352108225801136401, 4.09169692961501402302934776766, 4.70896218799320765539737235509, 6.35703521472593592285646322925, 6.76988851162287712729369638369, 7.61078180543088194909387998236, 9.039453479529042720771698331224, 9.565473522574294474518914798848, 10.00650582052408699040138304092

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