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Properties

Label 2-325-13.11-c2-0-27
Degree $2$
Conductor $325$
Sign $-0.364 + 0.931i$
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  + (0.810 − 3.02i)2-s + (1.62 + 2.80i)3-s + (−5.02 − 2.90i)4-s + (9.80 − 2.62i)6-s + (−1.22 − 4.59i)7-s + (−3.99 + 3.99i)8-s + (−0.752 + 1.30i)9-s + (7.33 + 1.96i)11-s − 18.8i·12-s + (5.20 − 11.9i)13-s − 14.8·14-s + (−2.76 − 4.79i)16-s + (14.1 + 8.17i)17-s + (3.33 + 3.33i)18-s + (−3.86 + 1.03i)19-s + ⋯
L(s)  = 1  + (0.405 − 1.51i)2-s + (0.540 + 0.935i)3-s + (−1.25 − 0.725i)4-s + (1.63 − 0.437i)6-s + (−0.175 − 0.655i)7-s + (−0.498 + 0.498i)8-s + (−0.0836 + 0.144i)9-s + (0.666 + 0.178i)11-s − 1.56i·12-s + (0.400 − 0.916i)13-s − 1.06·14-s + (−0.173 − 0.299i)16-s + (0.833 + 0.480i)17-s + (0.185 + 0.185i)18-s + (−0.203 + 0.0545i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.364 + 0.931i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.364 + 0.931i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.35337 - 1.98378i\)
\(L(\frac12)\) \(\approx\) \(1.35337 - 1.98378i\)
\(L(2)\) 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 \( 1 + (-5.20 + 11.9i)T \)
good2 \( 1 + (-0.810 + 3.02i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-1.62 - 2.80i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (1.22 + 4.59i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-7.33 - 1.96i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-14.1 - 8.17i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.86 - 1.03i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-10.8 + 6.26i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (21.9 + 38.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (32.0 + 32.0i)T + 961iT^{2} \)
37 \( 1 + (1.04 + 0.278i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (18.1 - 67.5i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-48.6 - 28.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-6.10 + 6.10i)T - 2.20e3iT^{2} \)
53 \( 1 - 50.1T + 2.80e3T^{2} \)
59 \( 1 + (-21.5 - 80.2i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (24.5 - 42.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.8 - 59.3i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (72.4 - 19.4i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (81.6 - 81.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 2.64T + 6.24e3T^{2} \)
83 \( 1 + (-45.5 - 45.5i)T + 6.88e3iT^{2} \)
89 \( 1 + (78.1 + 20.9i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-97.4 + 26.1i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−10.99812911885497433697005289193, −10.18121077622030735463641989146, −9.733191020291549700056254498424, −8.750551811417237519397092969630, −7.41877027029782427179873248398, −5.78565478766252232570449839276, −4.27082091294509247907451798600, −3.82722819785559354034670189453, −2.79037994206292488211659849116, −1.06885373438738456180231107240, 1.79312208872070954544678659582, 3.55740041398862300456332234099, 5.05132827369824436074949039583, 6.01490458727902760728179119563, 7.04320588728244053700076367277, 7.43470488283554225330096360087, 8.801874541583606513277553886947, 8.997801380825040585049383943390, 10.83445348553487419655473856122, 12.12203799175501254374899775725

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