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Properties

Label 2-325-13.2-c2-0-14
Degree $2$
Conductor $325$
Sign $0.999 - 0.0323i$
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  + (−0.490 − 0.131i)2-s + (1.23 + 2.14i)3-s + (−3.24 − 1.87i)4-s + (−0.325 − 1.21i)6-s + (−6.51 + 1.74i)7-s + (2.77 + 2.77i)8-s + (1.43 − 2.48i)9-s + (2.09 − 7.83i)11-s − 9.26i·12-s + (10.8 − 7.14i)13-s + 3.42·14-s + (6.48 + 11.2i)16-s + (26.5 + 15.3i)17-s + (−1.02 + 1.02i)18-s + (4.07 + 15.2i)19-s + ⋯
L(s)  = 1  + (−0.245 − 0.0656i)2-s + (0.412 + 0.715i)3-s + (−0.810 − 0.467i)4-s + (−0.0542 − 0.202i)6-s + (−0.930 + 0.249i)7-s + (0.347 + 0.347i)8-s + (0.159 − 0.275i)9-s + (0.190 − 0.712i)11-s − 0.772i·12-s + (0.835 − 0.549i)13-s + 0.244·14-s + (0.405 + 0.702i)16-s + (1.56 + 0.901i)17-s + (−0.0571 + 0.0571i)18-s + (0.214 + 0.800i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0323i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.999 - 0.0323i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ 0.999 - 0.0323i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.36008 + 0.0220345i\)
\(L(\frac12)\) \(\approx\) \(1.36008 + 0.0220345i\)
\(L(2)\) 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 \( 1 + (-10.8 + 7.14i)T \)
good2 \( 1 + (0.490 + 0.131i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (-1.23 - 2.14i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (6.51 - 1.74i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-2.09 + 7.83i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-26.5 - 15.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-4.07 - 15.2i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-14.9 + 8.60i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (15.3 + 26.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (9.21 - 9.21i)T - 961iT^{2} \)
37 \( 1 + (-1.41 + 5.29i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-35.2 - 9.44i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-44.5 - 25.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (13.3 + 13.3i)T + 2.20e3iT^{2} \)
53 \( 1 - 7.15T + 2.80e3T^{2} \)
59 \( 1 + (-80.4 + 21.5i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-57.4 + 99.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-57.0 - 15.2i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (10.3 + 38.7i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (49.5 + 49.5i)T + 5.32e3iT^{2} \)
79 \( 1 + 109.T + 6.24e3T^{2} \)
83 \( 1 + (84.6 - 84.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-10.2 + 38.3i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-1.93 - 7.22i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−11.07123115215916753366301898915, −10.11657252551294103811078588201, −9.654162118927639201699801152658, −8.765686243703973720271173709489, −7.990194573841314504054768929299, −6.22272711497871836919968589144, −5.53060010175879276545430140738, −3.95629874581069871419427356867, −3.31593843938564499851742266831, −0.965554143497363476741919257873, 1.07003491957795047250292776026, 2.93685862713250279264512225351, 4.07947489335914738445490543275, 5.40803630691111843514308529421, 7.11364553584430747513454019310, 7.32836073343195958380864971262, 8.617161982144699884227494781858, 9.395541809444105864921406231885, 10.14924165376627341861015686615, 11.53156713741076378051585405995

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