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Properties

Label 2-325-65.19-c2-0-33
Degree $2$
Conductor $325$
Sign $0.0530 + 0.998i$
Analytic cond. $8.85560$
Root an. cond. $2.97583$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.73 − 1.00i)2-s + (−2.83 − 1.63i)3-s + (9.46 − 5.46i)4-s + (−12.2 − 3.27i)6-s + (8.61 + 2.30i)7-s + (18.9 − 18.9i)8-s + (0.844 + 1.46i)9-s + (3.75 − 1.00i)11-s − 35.7·12-s + (−11.0 + 6.85i)13-s + 34.4·14-s + (29.9 − 51.7i)16-s + (−2.89 − 5.00i)17-s + (4.61 + 4.61i)18-s + (−16.7 − 4.48i)19-s + ⋯
L(s)  = 1  + (1.86 − 0.500i)2-s + (−0.943 − 0.544i)3-s + (2.36 − 1.36i)4-s + (−2.03 − 0.545i)6-s + (1.23 + 0.329i)7-s + (2.36 − 2.36i)8-s + (0.0938 + 0.162i)9-s + (0.341 − 0.0913i)11-s − 2.97·12-s + (−0.849 + 0.527i)13-s + 2.46·14-s + (1.86 − 3.23i)16-s + (−0.170 − 0.294i)17-s + (0.256 + 0.256i)18-s + (−0.881 − 0.236i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0530 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0530 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.0530 + 0.998i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ 0.0530 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.95353 - 2.80084i\)
\(L(\frac12)\) \(\approx\) \(2.95353 - 2.80084i\)
\(L(2)\) 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 \( 1 + (11.0 - 6.85i)T \)
good2 \( 1 + (-3.73 + 1.00i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (2.83 + 1.63i)T + (4.5 + 7.79i)T^{2} \)
7 \( 1 + (-8.61 - 2.30i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-3.75 + 1.00i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (2.89 + 5.00i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (16.7 + 4.48i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (4.89 - 8.48i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-7.28 + 12.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-26.8 + 26.8i)T - 961iT^{2} \)
37 \( 1 + (-10.9 - 40.8i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-11.2 - 42.0i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-29.7 - 51.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (50.4 - 50.4i)T - 2.20e3iT^{2} \)
53 \( 1 + 28.3iT - 2.80e3T^{2} \)
59 \( 1 + (-2.48 + 9.25i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-10.4 - 18.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (10.7 - 2.88i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (7.29 + 1.95i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (36.6 - 36.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 45.8T + 6.24e3T^{2} \)
83 \( 1 + (-42.4 - 42.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (-86.1 + 23.0i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-13.0 + 48.7i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−11.50755741224321751702822149902, −11.06372903823478335397286226130, −9.724609876115931274027495403010, −7.88073192294310757409775634559, −6.67088688788987641728290977081, −6.05825295559700466865298558087, −4.95955012228348900753746692749, −4.37725734591414503845227667131, −2.62501266070506714580692923821, −1.41037883701661209442858709549, 2.23318449092290479046889305964, 3.94965470020837632597443115217, 4.75757650379822915432125625480, 5.34935383661228755994476659019, 6.34875906579318792365418649461, 7.39413142558003482824941012312, 8.350670739596354861188025540468, 10.53599603861204565193544456263, 10.85698096704202699921648491741, 11.95813428027667857017371567882

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