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Properties

Label 2-325-65.2-c1-0-15
Degree $2$
Conductor $325$
Sign $0.887 + 0.460i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
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  + (−0.0621 − 0.107i)2-s + (2.43 + 0.652i)3-s + (0.992 − 1.71i)4-s + (−0.0810 − 0.302i)6-s + (−2.27 − 1.31i)7-s − 0.495·8-s + (2.90 + 1.67i)9-s + (0.780 − 2.91i)11-s + (3.53 − 3.53i)12-s + (3.34 + 1.34i)13-s + 0.325i·14-s + (−1.95 − 3.38i)16-s + (1.42 + 5.32i)17-s − 0.416i·18-s + (−3.13 + 0.840i)19-s + ⋯
L(s)  = 1  + (−0.0439 − 0.0760i)2-s + (1.40 + 0.376i)3-s + (0.496 − 0.859i)4-s + (−0.0330 − 0.123i)6-s + (−0.858 − 0.495i)7-s − 0.175·8-s + (0.968 + 0.559i)9-s + (0.235 − 0.878i)11-s + (1.02 − 1.02i)12-s + (0.928 + 0.372i)13-s + 0.0871i·14-s + (−0.488 − 0.846i)16-s + (0.346 + 1.29i)17-s − 0.0982i·18-s + (−0.719 + 0.192i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.887 + 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97290 - 0.480831i\)
\(L(\frac12)\) \(\approx\) \(1.97290 - 0.480831i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (-3.34 - 1.34i)T \)
good2 \( 1 + (0.0621 + 0.107i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-2.43 - 0.652i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (2.27 + 1.31i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.780 + 2.91i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.42 - 5.32i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.13 - 0.840i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.62 - 6.05i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.99 + 1.72i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.63 - 6.63i)T - 31iT^{2} \)
37 \( 1 + (1.89 - 1.09i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.78 - 2.35i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.89 + 0.775i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 2.26iT - 47T^{2} \)
53 \( 1 + (5.63 - 5.63i)T - 53iT^{2} \)
59 \( 1 + (3.33 + 12.4i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.79 + 4.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.96 + 6.86i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.12 - 15.4i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 6.95T + 73T^{2} \)
79 \( 1 - 5.90iT - 79T^{2} \)
83 \( 1 + 9.71iT - 83T^{2} \)
89 \( 1 + (-12.3 - 3.31i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.62 - 4.55i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.19583161174076573555545385678, −10.48490553425027806980319277054, −9.616298608638870010303069322500, −8.878049205553266187040390119351, −7.944114786720721967351358084393, −6.62428557422847636245811154729, −5.81889138147246795901413453668, −3.94050183650418655684684065271, −3.21858139836191739682843203487, −1.64931389190481658636666836205, 2.27889869759711552560589606063, 3.03162892855791677518456523691, 4.14147954396343018413078207643, 6.14773472480756166943131667648, 7.14019583131977984785182666509, 7.87485160027641332989687489074, 8.868276087403466790717775115087, 9.371645740553936593246527165156, 10.72244940367283069877902072009, 11.98840093574112465473053718405

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