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Properties

Label 2-325-13.6-c2-0-0
Degree $2$
Conductor $325$
Sign $-0.648 + 0.760i$
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  + (0.0532 + 0.198i)2-s + (−2.75 + 4.76i)3-s + (3.42 − 1.97i)4-s + (−1.09 − 0.292i)6-s + (1.16 − 4.33i)7-s + (1.15 + 1.15i)8-s + (−10.6 − 18.4i)9-s + (−18.8 + 5.05i)11-s + 21.7i·12-s + (−8.04 + 10.2i)13-s + 0.923·14-s + (7.74 − 13.4i)16-s + (−10.6 + 6.15i)17-s + (3.09 − 3.09i)18-s + (−11.7 − 3.15i)19-s + ⋯
L(s)  = 1  + (0.0266 + 0.0993i)2-s + (−0.916 + 1.58i)3-s + (0.856 − 0.494i)4-s + (−0.182 − 0.0488i)6-s + (0.165 − 0.619i)7-s + (0.144 + 0.144i)8-s + (−1.18 − 2.04i)9-s + (−1.71 + 0.459i)11-s + 1.81i·12-s + (−0.619 + 0.785i)13-s + 0.0659·14-s + (0.484 − 0.838i)16-s + (−0.627 + 0.362i)17-s + (0.171 − 0.171i)18-s + (−0.619 − 0.165i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.760i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.648 + 0.760i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.648 + 0.760i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0387320 - 0.0839379i\)
\(L(\frac12)\) \(\approx\) \(0.0387320 - 0.0839379i\)
\(L(2)\) 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 \( 1 + (8.04 - 10.2i)T \)
good2 \( 1 + (-0.0532 - 0.198i)T + (-3.46 + 2i)T^{2} \)
3 \( 1 + (2.75 - 4.76i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + (-1.16 + 4.33i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (18.8 - 5.05i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (10.6 - 6.15i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (11.7 + 3.15i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (28.0 + 16.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (0.962 - 1.66i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-25.6 + 25.6i)T - 961iT^{2} \)
37 \( 1 + (26.8 - 7.20i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-10.3 - 38.7i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (0.993 - 0.573i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (31.6 + 31.6i)T + 2.20e3iT^{2} \)
53 \( 1 + 26.6T + 2.80e3T^{2} \)
59 \( 1 + (1.92 - 7.16i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-21.1 - 36.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (2.01 + 7.53i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-26.8 - 7.19i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-77.2 - 77.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 77.7T + 6.24e3T^{2} \)
83 \( 1 + (7.65 - 7.65i)T - 6.88e3iT^{2} \)
89 \( 1 + (149. - 40.1i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (141. + 37.9i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−11.56083200665958988726894552921, −10.88971665987789065698931600800, −10.20049471529622759424643358282, −9.798312046726908994347075782050, −8.230297615638795297079406728355, −6.91411661391292296418118221650, −5.95351689162337015465125299099, −4.93490395753108470738838730038, −4.23089674308609529239715239191, −2.46319603445626659460547014034, 0.04232239336771460278720027544, 1.98733408586560751663142695855, 2.79159003864142659436484168860, 5.21648231196178809985732008727, 5.94571408515582459130445743130, 6.94673951896502578465514518734, 7.83982832604670750195065485466, 8.291415245995119384954056511442, 10.36840979429208911274582999240, 11.04026913006097933007292850463

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