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Properties

Label 2-325-65.54-c2-0-13
Degree $2$
Conductor $325$
Sign $-0.167 - 0.985i$
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  + (−0.974 + 3.63i)2-s + (−4.47 + 2.58i)3-s + (−8.81 − 5.09i)4-s + (−5.03 − 18.8i)6-s + (−0.239 − 0.893i)7-s + (16.4 − 16.4i)8-s + (8.86 − 15.3i)9-s + (−0.879 + 3.28i)11-s + 52.6·12-s + (−4.93 + 12.0i)13-s + 3.48·14-s + (23.4 + 40.6i)16-s + (5.69 − 9.86i)17-s + (47.2 + 47.2i)18-s + (−3.16 − 11.7i)19-s + ⋯
L(s)  = 1  + (−0.487 + 1.81i)2-s + (−1.49 + 0.861i)3-s + (−2.20 − 1.27i)4-s + (−0.839 − 3.13i)6-s + (−0.0341 − 0.127i)7-s + (2.05 − 2.05i)8-s + (0.985 − 1.70i)9-s + (−0.0799 + 0.298i)11-s + 4.38·12-s + (−0.379 + 0.925i)13-s + 0.248·14-s + (1.46 + 2.54i)16-s + (0.334 − 0.580i)17-s + (2.62 + 2.62i)18-s + (−0.166 − 0.620i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.167 - 0.985i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.167 - 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.275051 + 0.325853i\)
\(L(\frac12)\) \(\approx\) \(0.275051 + 0.325853i\)
\(L(2)\) 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 \( 1 + (4.93 - 12.0i)T \)
good2 \( 1 + (0.974 - 3.63i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (4.47 - 2.58i)T + (4.5 - 7.79i)T^{2} \)
7 \( 1 + (0.239 + 0.893i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (0.879 - 3.28i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-5.69 + 9.86i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (3.16 + 11.7i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (19.6 + 33.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-5.86 - 10.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-18.4 + 18.4i)T - 961iT^{2} \)
37 \( 1 + (-14.5 - 3.90i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-9.84 - 2.63i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (16.7 - 29.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (40.8 - 40.8i)T - 2.20e3iT^{2} \)
53 \( 1 - 21.3iT - 2.80e3T^{2} \)
59 \( 1 + (-10.2 + 2.73i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (47.5 - 82.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-6.90 + 25.7i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (10.2 + 38.0i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-89.2 + 89.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 8.13T + 6.24e3T^{2} \)
83 \( 1 + (46.6 + 46.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (10.0 - 37.6i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-135. + 36.4i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.47433369679853975453141756125, −10.32682579885184403250976135613, −9.732460202989487474570185676913, −8.853015478653998842344264438055, −7.54112973816838777593076135814, −6.57154573052766960542652337382, −6.01360175645095053739255306952, −4.78135205682578797364786806915, −4.44995571789483588049961734696, −0.44483077333465743437998969443, 0.828662239180094438942836074727, 1.99657303052132550307391622162, 3.57237184954059583571209179543, 5.04144210642509316093657655850, 5.97749670053537968653180004697, 7.56400444948316845560249251783, 8.375183966086068021017768343576, 9.886981868673389186021542722656, 10.39620009946310285093436635259, 11.30893995005833597237190438137

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