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Properties

Label 2-325-65.59-c2-0-3
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: $\chi_{325} (124, \cdot )$
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

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

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