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Properties

Label 2-840-35.17-c1-0-23
Degree $2$
Conductor $840$
Sign $-0.379 + 0.925i$
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 + (−0.554 − 2.16i)5-s + (0.714 − 2.54i)7-s + (0.866 − 0.499i)9-s + (−1.60 + 2.77i)11-s + (−1.82 − 1.82i)13-s + (−1.09 − 1.94i)15-s + (−0.708 − 2.64i)17-s + (−3.10 − 5.37i)19-s + (0.0305 − 2.64i)21-s + (8.10 + 2.17i)23-s + (−4.38 + 2.40i)25-s + (0.707 − 0.707i)27-s + 0.962i·29-s + (−5.02 − 2.89i)31-s + ⋯
L(s)  = 1  + (0.557 − 0.149i)3-s + (−0.247 − 0.968i)5-s + (0.269 − 0.962i)7-s + (0.288 − 0.166i)9-s + (−0.483 + 0.837i)11-s + (−0.506 − 0.506i)13-s + (−0.283 − 0.503i)15-s + (−0.171 − 0.641i)17-s + (−0.711 − 1.23i)19-s + (0.00667 − 0.577i)21-s + (1.69 + 0.452i)23-s + (−0.877 + 0.480i)25-s + (0.136 − 0.136i)27-s + 0.178i·29-s + (−0.901 − 0.520i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.379 + 0.925i$
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: \((2,\ 840,\ (\ :1/2),\ -0.379 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.843179 - 1.25781i\)
\(L(\frac12)\) \(\approx\) \(0.843179 - 1.25781i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.554 + 2.16i)T \)
7 \( 1 + (-0.714 + 2.54i)T \)
good11 \( 1 + (1.60 - 2.77i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.82 + 1.82i)T + 13iT^{2} \)
17 \( 1 + (0.708 + 2.64i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.10 + 5.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-8.10 - 2.17i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.962iT - 29T^{2} \)
31 \( 1 + (5.02 + 2.89i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.153 + 0.572i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.31iT - 41T^{2} \)
43 \( 1 + (2.74 - 2.74i)T - 43iT^{2} \)
47 \( 1 + (-2.49 - 0.668i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.34 + 8.73i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.51 + 6.09i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.97 - 2.29i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.28 + 2.48i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 + (3.95 - 1.06i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-14.9 + 8.64i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.31 - 2.31i)T + 83iT^{2} \)
89 \( 1 + (2.47 + 4.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.879 + 0.879i)T - 97iT^{2} \)
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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

−9.708741724900181387270957768946, −9.160412402962177162720598338081, −8.163444579490754224488298989146, −7.44748750542424190718131618345, −6.82867530702733760305501266542, −5.05367033356637224094819507687, −4.72789778475848797424150833477, −3.48997756185298338830108489191, −2.15603660137342459542288550236, −0.67970168401559188178419260280, 2.04417983609269021041277885109, 2.95280924445027233644399102286, 3.91071802038233720163412162612, 5.19700964840418281567599599496, 6.16616477741178535286190025558, 7.07539044505722736126739420033, 8.084748269258661917436787440475, 8.669334956038155696591044045808, 9.551374421231122485372018133138, 10.66142766276298788227982472975

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