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Properties

Label 2-325-65.54-c2-0-2
Degree $2$
Conductor $325$
Sign $0.481 - 0.876i$
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  + (0.683 − 2.54i)2-s + (−3.75 + 2.17i)3-s + (−2.57 − 1.48i)4-s + (2.96 + 11.0i)6-s + (−2.17 − 8.11i)7-s + (1.92 − 1.92i)8-s + (4.91 − 8.51i)9-s + (−3.57 + 13.3i)11-s + 12.8·12-s + (−11.3 + 6.35i)13-s − 22.1·14-s + (−9.53 − 16.5i)16-s + (−4.27 + 7.39i)17-s + (−18.3 − 18.3i)18-s + (3.60 + 13.4i)19-s + ⋯
L(s)  = 1  + (0.341 − 1.27i)2-s + (−1.25 + 0.723i)3-s + (−0.642 − 0.371i)4-s + (0.494 + 1.84i)6-s + (−0.310 − 1.15i)7-s + (0.240 − 0.240i)8-s + (0.546 − 0.946i)9-s + (−0.324 + 1.21i)11-s + 1.07·12-s + (−0.872 + 0.488i)13-s − 1.58·14-s + (−0.595 − 1.03i)16-s + (−0.251 + 0.435i)17-s + (−1.02 − 1.02i)18-s + (0.189 + 0.708i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.481 - 0.876i$
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: \((2,\ 325,\ (\ :1),\ 0.481 - 0.876i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.467810 + 0.276663i\)
\(L(\frac12)\) \(\approx\) \(0.467810 + 0.276663i\)
\(L(2)\) 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 \( 1 + (11.3 - 6.35i)T \)
good2 \( 1 + (-0.683 + 2.54i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (3.75 - 2.17i)T + (4.5 - 7.79i)T^{2} \)
7 \( 1 + (2.17 + 8.11i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (3.57 - 13.3i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (4.27 - 7.39i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.60 - 13.4i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-10.5 - 18.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-27.0 - 46.8i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (15.3 - 15.3i)T - 961iT^{2} \)
37 \( 1 + (4.20 + 1.12i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (4.73 + 1.26i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (35.1 - 60.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (22.4 - 22.4i)T - 2.20e3iT^{2} \)
53 \( 1 + 50.1iT - 2.80e3T^{2} \)
59 \( 1 + (99.2 - 26.6i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-20.1 + 34.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (8.14 - 30.4i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (36.0 + 134. i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (43.7 - 43.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 20.8T + 6.24e3T^{2} \)
83 \( 1 + (-50.3 - 50.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (-28.5 + 106. i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (3.90 - 1.04i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.44798408816525027490185485719, −10.62534312685867290814511892766, −10.16410558031019197581836155421, −9.503777229460597534426865164144, −7.46469057559869790938052367336, −6.64937290470647325397932756214, −4.99174493288071192934645612039, −4.48685035511938000493446740388, −3.35337446553697880390201582159, −1.52992237741866817490502892219, 0.26736146925246473085526201123, 2.58661893541018957717540696902, 4.87358009815235800607675599206, 5.59815423412744102301962539129, 6.21890948238199573496323147077, 7.02493170072270499765725796659, 8.047428655954582342149846704800, 9.019487593960246828530221967452, 10.53169615335635034391211415171, 11.48405793451964906008252161146

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