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Properties

Label 2-325-13.11-c2-0-30
Degree $2$
Conductor $325$
Sign $-0.862 + 0.505i$
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  + (−0.534 + 1.99i)2-s + (−2.25 − 3.91i)3-s + (−0.231 − 0.133i)4-s + (9.01 − 2.41i)6-s + (−1.95 − 7.29i)7-s + (−5.45 + 5.45i)8-s + (−5.70 + 9.88i)9-s + (5.50 + 1.47i)11-s + 1.20i·12-s + (11.4 − 6.13i)13-s + 15.6·14-s + (−8.49 − 14.7i)16-s + (−3.81 − 2.20i)17-s + (−16.6 − 16.6i)18-s + (−28.2 + 7.57i)19-s + ⋯
L(s)  = 1  + (−0.267 + 0.997i)2-s + (−0.753 − 1.30i)3-s + (−0.0579 − 0.0334i)4-s + (1.50 − 0.402i)6-s + (−0.279 − 1.04i)7-s + (−0.681 + 0.681i)8-s + (−0.634 + 1.09i)9-s + (0.500 + 0.134i)11-s + 0.100i·12-s + (0.881 − 0.471i)13-s + 1.11·14-s + (−0.531 − 0.920i)16-s + (−0.224 − 0.129i)17-s + (−0.926 − 0.926i)18-s + (−1.48 + 0.398i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.505i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.862 + 0.505i$
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: \((2,\ 325,\ (\ :1),\ -0.862 + 0.505i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0648193 - 0.238830i\)
\(L(\frac12)\) \(\approx\) \(0.0648193 - 0.238830i\)
\(L(2)\) 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 \( 1 + (-11.4 + 6.13i)T \)
good2 \( 1 + (0.534 - 1.99i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (2.25 + 3.91i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (1.95 + 7.29i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-5.50 - 1.47i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (3.81 + 2.20i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (28.2 - 7.57i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (33.6 - 19.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (1.49 + 2.59i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (11.6 + 11.6i)T + 961iT^{2} \)
37 \( 1 + (39.0 + 10.4i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (4.81 - 17.9i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (36.2 + 20.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.03 - 1.03i)T - 2.20e3iT^{2} \)
53 \( 1 - 29.2T + 2.80e3T^{2} \)
59 \( 1 + (18.7 + 70.0i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-34.5 + 59.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (29.6 - 110. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (84.3 - 22.5i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (55.2 - 55.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 27.1T + 6.24e3T^{2} \)
83 \( 1 + (-74.2 - 74.2i)T + 6.88e3iT^{2} \)
89 \( 1 + (-23.1 - 6.18i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-177. + 47.6i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.19917032058802748959968727254, −10.15137380236251314295084770343, −8.626416191290879984728920855989, −7.81720830237005455766830898816, −7.00535116787949634748033349250, −6.38281939809095849521086234642, −5.65323586563295280131431696452, −3.84111481500220749805465937910, −1.79852729091665033520060210689, −0.13126192549013080002665006054, 2.03874376891600744658235801457, 3.52170853697059926531810650264, 4.47138122334798178411607378630, 5.92874108191149367987826212505, 6.44722228073150945212564860530, 8.763423909551236086521232472571, 9.103569686054712271748123563635, 10.36506617123617975609777070179, 10.61086430989246892768221253608, 11.73365900204861391983052691808

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