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Properties

Label 2-840-35.3-c1-0-13
Degree $2$
Conductor $840$
Sign $0.955 + 0.295i$
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  + (0.258 + 0.965i)3-s + (−1.93 + 1.12i)5-s + (−1.60 − 2.10i)7-s + (−0.866 + 0.499i)9-s + (2.59 − 4.49i)11-s + (0.393 − 0.393i)13-s + (−1.58 − 1.57i)15-s + (4.10 − 1.10i)17-s + (2.32 + 4.02i)19-s + (1.62 − 2.09i)21-s + (0.979 − 3.65i)23-s + (2.48 − 4.33i)25-s + (−0.707 − 0.707i)27-s − 1.14i·29-s + (9.03 + 5.21i)31-s + ⋯
L(s)  = 1  + (0.149 + 0.557i)3-s + (−0.865 + 0.501i)5-s + (−0.604 − 0.796i)7-s + (−0.288 + 0.166i)9-s + (0.781 − 1.35i)11-s + (0.109 − 0.109i)13-s + (−0.408 − 0.407i)15-s + (0.996 − 0.267i)17-s + (0.533 + 0.923i)19-s + (0.353 − 0.456i)21-s + (0.204 − 0.762i)23-s + (0.497 − 0.867i)25-s + (−0.136 − 0.136i)27-s − 0.211i·29-s + (1.62 + 0.937i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.955 + 0.295i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.955 + 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29752 - 0.195915i\)
\(L(\frac12)\) \(\approx\) \(1.29752 - 0.195915i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (1.93 - 1.12i)T \)
7 \( 1 + (1.60 + 2.10i)T \)
good11 \( 1 + (-2.59 + 4.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.393 + 0.393i)T - 13iT^{2} \)
17 \( 1 + (-4.10 + 1.10i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.32 - 4.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.979 + 3.65i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 1.14iT - 29T^{2} \)
31 \( 1 + (-9.03 - 5.21i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.54 + 1.75i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 3.89iT - 41T^{2} \)
43 \( 1 + (-8.49 - 8.49i)T + 43iT^{2} \)
47 \( 1 + (-2.37 + 8.87i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.74 - 1.00i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.64 + 2.85i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.21 + 1.85i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.26 + 8.44i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + (2.84 + 10.6i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-14.8 + 8.57i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.465 + 0.465i)T - 83iT^{2} \)
89 \( 1 + (-1.44 - 2.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.58 - 3.58i)T + 97iT^{2} \)
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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

−10.39608533250908969719820091335, −9.362144289180160721358721471124, −8.420710978716986583090624980178, −7.70203401548314704445027860295, −6.69802587906932871808452448511, −5.89200866173129842658611329691, −4.53888114358075864289328584194, −3.50759196527229399380625107717, −3.17465826786129322980070476885, −0.77693343416722606095015029268, 1.20663070841444412157755894503, 2.67594843700333457625787758631, 3.81043338249327075321958029503, 4.86808791677303635283814399693, 5.93127649395940981837031424294, 6.99721247040464221325492889917, 7.57501347189121063884734317933, 8.601658251390555936066520662349, 9.292911063871063259776756202604, 10.00405232655235667023293938259

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