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Properties

Label 2-325-13.10-c1-0-15
Degree $2$
Conductor $325$
Sign $0.879 + 0.475i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.27 − 1.31i)2-s + (1.34 + 2.33i)3-s + (2.46 − 4.26i)4-s + (6.15 + 3.55i)6-s + (−2.37 − 1.37i)7-s − 7.71i·8-s + (−2.14 + 3.71i)9-s + (−1.59 + 0.920i)11-s + 13.3·12-s + (−3.60 + 0.149i)13-s − 7.21·14-s + (−5.22 − 9.04i)16-s + (−1.90 + 3.30i)17-s + 11.2i·18-s + (4.91 + 2.83i)19-s + ⋯
L(s)  = 1  + (1.61 − 0.930i)2-s + (0.779 + 1.34i)3-s + (1.23 − 2.13i)4-s + (2.51 + 1.45i)6-s + (−0.897 − 0.517i)7-s − 2.72i·8-s + (−0.714 + 1.23i)9-s + (−0.480 + 0.277i)11-s + 3.84·12-s + (−0.999 + 0.0413i)13-s − 1.92·14-s + (−1.30 − 2.26i)16-s + (−0.462 + 0.801i)17-s + 2.65i·18-s + (1.12 + 0.650i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.29636 - 0.833059i\)
\(L(\frac12)\) \(\approx\) \(3.29636 - 0.833059i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 - 0.149i)T \)
good2 \( 1 + (-2.27 + 1.31i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.34 - 2.33i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.37 + 1.37i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.59 - 0.920i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.90 - 3.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.91 - 2.83i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0696 + 0.120i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.583 + 1.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.69iT - 31T^{2} \)
37 \( 1 + (-2.13 + 1.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.01 + 4.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.04 - 1.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.66iT - 47T^{2} \)
53 \( 1 + 9.97T + 53T^{2} \)
59 \( 1 + (-2.19 - 1.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.17 + 7.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.18 + 2.99i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.58 + 4.38i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 + 9.64T + 79T^{2} \)
83 \( 1 + 15.6iT - 83T^{2} \)
89 \( 1 + (-2.91 + 1.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.271 - 0.156i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.55673815916855880944373680085, −10.55452420423087770666896417977, −9.970974977352542526212749337551, −9.389477176053971767411866594845, −7.60925043031197940748505518857, −6.15459013288076274064372275682, −5.02336827494401496888324740650, −4.14624038116731878918830723136, −3.40384082129866183431502332284, −2.41577245374162864591602490615, 2.61465306359450860401866787332, 3.10293956189162759042713139553, 4.85943099372170031280554900507, 5.89865928250134793929034420755, 6.93916824638702496156117167302, 7.33713575525911192232751455222, 8.349493662033536100433854107516, 9.479257965628576038772165109201, 11.44633930629714899442729756020, 12.30299666426495048728128359094

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