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Properties

Label 2-325-13.2-c2-0-13
Degree $2$
Conductor $325$
Sign $0.946 + 0.322i$
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  + (−2.79 − 0.749i)2-s + (−0.266 − 0.461i)3-s + (3.80 + 2.19i)4-s + (0.399 + 1.49i)6-s + (10.0 − 2.69i)7-s + (−0.798 − 0.798i)8-s + (4.35 − 7.54i)9-s + (−3.30 + 12.3i)11-s − 2.34i·12-s + (−12.6 + 2.84i)13-s − 30.1·14-s + (−7.14 − 12.3i)16-s + (19.2 + 11.1i)17-s + (−17.8 + 17.8i)18-s + (4.52 + 16.8i)19-s + ⋯
L(s)  = 1  + (−1.39 − 0.374i)2-s + (−0.0888 − 0.153i)3-s + (0.950 + 0.548i)4-s + (0.0666 + 0.248i)6-s + (1.43 − 0.384i)7-s + (−0.0998 − 0.0998i)8-s + (0.484 − 0.838i)9-s + (−0.300 + 1.12i)11-s − 0.195i·12-s + (−0.975 + 0.218i)13-s − 2.15·14-s + (−0.446 − 0.773i)16-s + (1.13 + 0.653i)17-s + (−0.991 + 0.991i)18-s + (0.237 + 0.888i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.946 + 0.322i$
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: \((2,\ 325,\ (\ :1),\ 0.946 + 0.322i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.900086 - 0.149231i\)
\(L(\frac12)\) \(\approx\) \(0.900086 - 0.149231i\)
\(L(2)\) 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 \( 1 + (12.6 - 2.84i)T \)
good2 \( 1 + (2.79 + 0.749i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (0.266 + 0.461i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (-10.0 + 2.69i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (3.30 - 12.3i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-19.2 - 11.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-4.52 - 16.8i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (7.30 - 4.21i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-15.3 - 26.5i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-13.1 + 13.1i)T - 961iT^{2} \)
37 \( 1 + (-13.2 + 49.4i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-29.7 - 7.97i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (0.921 + 0.531i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (9.18 + 9.18i)T + 2.20e3iT^{2} \)
53 \( 1 - 43.6T + 2.80e3T^{2} \)
59 \( 1 + (-97.0 + 26.0i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-6.91 + 11.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-97.0 - 25.9i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (0.479 + 1.78i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-74.7 - 74.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 132.T + 6.24e3T^{2} \)
83 \( 1 + (53.8 - 53.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (-27.9 + 104. i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (34.9 + 130. i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−11.17052139426779320451381066760, −10.03083956737399147296814862073, −9.826727649548731327404715034396, −8.482187365365597049918458425616, −7.66771131529977678056576244318, −7.11756635626189170390004945088, −5.33268111858859545858282074818, −4.14689034641131513758703110939, −2.09434628728981005889713038942, −1.10146575519153114427535855496, 0.923581581660854597786733498517, 2.46949559738579055486671108483, 4.63221660469782586783611926853, 5.51302271565082067281545439533, 7.07421396513001528406250028137, 8.008827066051558379848060989609, 8.307635065679793125795824019698, 9.569926898333049298149127394784, 10.32124950061172660280123154443, 11.17189599993795843498373260527

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