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Properties

Label 2-325-65.64-c3-0-8
Degree $2$
Conductor $325$
Sign $0.541 - 0.840i$
Analytic cond. $19.1756$
Root an. cond. $4.37899$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.742·2-s − 5.13i·3-s − 7.44·4-s − 3.81i·6-s − 9.97·7-s − 11.4·8-s + 0.594·9-s + 6.66i·11-s + 38.2i·12-s + (−46.5 − 5.10i)13-s − 7.40·14-s + 51.0·16-s − 1.15i·17-s + 0.441·18-s + 48.3i·19-s + ⋯
L(s)  = 1  + 0.262·2-s − 0.988i·3-s − 0.931·4-s − 0.259i·6-s − 0.538·7-s − 0.506·8-s + 0.0220·9-s + 0.182i·11-s + 0.920i·12-s + (−0.994 − 0.108i)13-s − 0.141·14-s + 0.798·16-s − 0.0164i·17-s + 0.00577·18-s + 0.583i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.541 - 0.840i$
Analytic conductor: \(19.1756\)
Root analytic conductor: \(4.37899\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :3/2),\ 0.541 - 0.840i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8550934213\)
\(L(\frac12)\) \(\approx\) \(0.8550934213\)
\(L(\frac{5}{2})\) 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 \( 1 + (46.5 + 5.10i)T \)
good2 \( 1 - 0.742T + 8T^{2} \)
3 \( 1 + 5.13iT - 27T^{2} \)
7 \( 1 + 9.97T + 343T^{2} \)
11 \( 1 - 6.66iT - 1.33e3T^{2} \)
17 \( 1 + 1.15iT - 4.91e3T^{2} \)
19 \( 1 - 48.3iT - 6.85e3T^{2} \)
23 \( 1 - 125. iT - 1.21e4T^{2} \)
29 \( 1 - 139.T + 2.43e4T^{2} \)
31 \( 1 - 195. iT - 2.97e4T^{2} \)
37 \( 1 - 174.T + 5.06e4T^{2} \)
41 \( 1 - 452. iT - 6.89e4T^{2} \)
43 \( 1 + 54.5iT - 7.95e4T^{2} \)
47 \( 1 - 369.T + 1.03e5T^{2} \)
53 \( 1 + 4.19iT - 1.48e5T^{2} \)
59 \( 1 + 169. iT - 2.05e5T^{2} \)
61 \( 1 - 271.T + 2.26e5T^{2} \)
67 \( 1 + 54.6T + 3.00e5T^{2} \)
71 \( 1 - 883. iT - 3.57e5T^{2} \)
73 \( 1 + 613.T + 3.89e5T^{2} \)
79 \( 1 + 425.T + 4.93e5T^{2} \)
83 \( 1 + 680.T + 5.71e5T^{2} \)
89 \( 1 - 943. iT - 7.04e5T^{2} \)
97 \( 1 + 1.81e3T + 9.12e5T^{2} \)
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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

−11.71137449244858918588195744919, −10.11488777082046883779585704973, −9.575931415221303966840583410217, −8.354282405462586597838894115115, −7.48328648956029882279016354925, −6.50959485166434403658259048938, −5.38608163082979434192347893562, −4.24583501977202538816720433479, −2.87813965575833789482074598897, −1.20138469624739790102201642636, 0.32972245259846679840215035406, 2.81836832896434153655907618913, 4.10217936153156195964957942571, 4.71385368747047714291403332899, 5.81354268750543471408754297072, 7.13967847844381127202211408682, 8.489985150725412072357212893697, 9.373572346079579855993933992738, 9.937433635064848716963412199876, 10.78192867565727496064276768669

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