Properties

Label 2-325-5.4-c3-0-20
Degree $2$
Conductor $325$
Sign $-0.894 - 0.447i$
Analytic cond. $19.1756$
Root an. cond. $4.37899$
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  + 2.56i·2-s + 3.68i·3-s + 1.43·4-s − 9.43·6-s + 18.1i·7-s + 24.1i·8-s + 13.4·9-s + 64.7·11-s + 5.30i·12-s + 13i·13-s − 46.5·14-s − 50.4·16-s − 25.5i·17-s + 34.3i·18-s + 107.·19-s + ⋯
L(s)  = 1  + 0.905i·2-s + 0.709i·3-s + 0.179·4-s − 0.642·6-s + 0.981i·7-s + 1.06i·8-s + 0.497·9-s + 1.77·11-s + 0.127i·12-s + 0.277i·13-s − 0.888·14-s − 0.787·16-s − 0.364i·17-s + 0.450i·18-s + 1.30·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(19.1756\)
Root analytic conductor: \(4.37899\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.479952775\)
\(L(\frac12)\) \(\approx\) \(2.479952775\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 - 13iT \)
good2 \( 1 - 2.56iT - 8T^{2} \)
3 \( 1 - 3.68iT - 27T^{2} \)
7 \( 1 - 18.1iT - 343T^{2} \)
11 \( 1 - 64.7T + 1.33e3T^{2} \)
17 \( 1 + 25.5iT - 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
23 \( 1 + 73.2iT - 1.21e4T^{2} \)
29 \( 1 + 175.T + 2.43e4T^{2} \)
31 \( 1 + 113.T + 2.97e4T^{2} \)
37 \( 1 - 114. iT - 5.06e4T^{2} \)
41 \( 1 + 69.6T + 6.89e4T^{2} \)
43 \( 1 + 438. iT - 7.95e4T^{2} \)
47 \( 1 + 31.9iT - 1.03e5T^{2} \)
53 \( 1 + 2.84iT - 1.48e5T^{2} \)
59 \( 1 + 71.6T + 2.05e5T^{2} \)
61 \( 1 + 920.T + 2.26e5T^{2} \)
67 \( 1 + 444. iT - 3.00e5T^{2} \)
71 \( 1 + 541.T + 3.57e5T^{2} \)
73 \( 1 + 764. iT - 3.89e5T^{2} \)
79 \( 1 - 421.T + 4.93e5T^{2} \)
83 \( 1 + 603. iT - 5.71e5T^{2} \)
89 \( 1 - 1.15e3T + 7.04e5T^{2} \)
97 \( 1 - 583. iT - 9.12e5T^{2} \)
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

−11.72136503609301718177513161390, −10.62753553101702144082288535148, −9.249783731033317020308087182306, −9.040667915827122377056101448542, −7.56961256098435173752296635982, −6.71758275121854846080532643478, −5.76390430970749247955223197594, −4.76779479718188982490420179830, −3.46048721376878680794274874190, −1.78253068736606955519741748194, 1.01438531445432514586042981298, 1.64359821011770259688267907668, 3.39846612707465757554114803862, 4.19353380197353844081808288892, 6.08596167276612648476747002950, 7.08741802148926598695932822741, 7.56839771430920734485331135983, 9.273423850468515179954713811614, 9.931809113758408879000518478340, 11.00725374196556733210250762955

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