↑

Properties

Label 2-84-7.4-c3-0-1
Degree $2$
Conductor $84$
Sign $0.959 - 0.281i$
Analytic cond. $4.95616$
Root an. cond. $2.22624$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 2.59i)3-s + (6.41 + 11.1i)5-s + (6.32 + 17.4i)7-s + (−4.5 − 7.79i)9-s + (18.4 − 31.8i)11-s + 87.1·13-s + 38.4·15-s + (−51.2 + 88.8i)17-s + (−47.9 − 82.9i)19-s + (54.7 + 9.67i)21-s + (48 + 83.1i)23-s + (−19.7 + 34.1i)25-s − 27·27-s − 212.·29-s + (79.6 − 137. i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.573 + 0.993i)5-s + (0.341 + 0.939i)7-s + (−0.166 − 0.288i)9-s + (0.504 − 0.874i)11-s + 1.85·13-s + 0.662·15-s + (−0.731 + 1.26i)17-s + (−0.578 − 1.00i)19-s + (0.568 + 0.100i)21-s + (0.435 + 0.753i)23-s + (−0.157 + 0.273i)25-s − 0.192·27-s − 1.35·29-s + (0.461 − 0.799i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.959 - 0.281i$
Analytic conductor: \(4.95616\)
Root analytic conductor: \(2.22624\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :3/2),\ 0.959 - 0.281i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.86751 + 0.268072i\)
\(L(\frac12)\) \(\approx\) \(1.86751 + 0.268072i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 + (-6.32 - 17.4i)T \)
good5 \( 1 + (-6.41 - 11.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-18.4 + 31.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 87.1T + 2.19e3T^{2} \)
17 \( 1 + (51.2 - 88.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (47.9 + 82.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-48 - 83.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 212.T + 2.43e4T^{2} \)
31 \( 1 + (-79.6 + 137. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (64.3 + 111. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 298.T + 6.89e4T^{2} \)
43 \( 1 + 33.3T + 7.95e4T^{2} \)
47 \( 1 + (135. + 234. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (224. - 388. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-334. + 578. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-121. - 211. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-167. + 290. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 339.T + 3.57e5T^{2} \)
73 \( 1 + (-459. + 795. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-68.1 - 118. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 287.T + 5.71e5T^{2} \)
89 \( 1 + (80.9 + 140. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 182.T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.71273358807759770242188441942, −13.06772660323382855085444514488, −11.40912511490236574004651641582, −10.86192158221473769576061011572, −9.093586811611967020738344226582, −8.339340041986204052957048686325, −6.57036966599470673810576772288, −5.90567545470002776879963500318, −3.48676924122324109927420705685, −1.90922406168941641732978896319, 1.45085552826094445501563705851, 3.92245742892394119045933587481, 5.03280974846418709692542862397, 6.69675445970896960112589091125, 8.332515498973622113980400139460, 9.218041210590957344530821352674, 10.33410358895498125240829225362, 11.43947424302314608191921697488, 12.94832394226332258820254743848, 13.65781771604437188712735995872

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