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Properties

Label 2-325-65.42-c0-0-1
Degree $2$
Conductor $325$
Sign $-0.0235 + 0.999i$
Analytic cond. $0.162196$
Root an. cond. $0.402735$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.499 + 0.866i)6-s + (0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.5 − 0.866i)11-s + (−0.707 + 0.707i)13-s − i·14-s + (−0.5 − 0.866i)16-s + (−0.965 + 0.258i)17-s + (0.866 + 0.5i)19-s − 21-s + (−0.965 + 0.258i)22-s + (0.965 + 0.258i)23-s + (−0.866 + 0.499i)24-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.499 + 0.866i)6-s + (0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.5 − 0.866i)11-s + (−0.707 + 0.707i)13-s − i·14-s + (−0.5 − 0.866i)16-s + (−0.965 + 0.258i)17-s + (0.866 + 0.5i)19-s − 21-s + (−0.965 + 0.258i)22-s + (0.965 + 0.258i)23-s + (−0.866 + 0.499i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0235 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0235 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.0235 + 0.999i$
Analytic conductor: \(0.162196\)
Root analytic conductor: \(0.402735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :0),\ -0.0235 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7717669665\)
\(L(\frac12)\) \(\approx\) \(0.7717669665\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
3 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
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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

−11.33827186063855993371792440232, −11.21855710908900607333189015821, −10.23218693661052069316603286708, −8.920781123549681211874269452494, −7.63640360980038919049684516027, −6.74595036114229929058325722945, −5.45502813956234140769918045969, −4.50121901249456491606044007286, −3.05729677438406405111249475696, −1.50316105702453525715934316855, 2.32021644006484557558852550951, 4.89396078126619507131035953431, 4.97428864036434284197320686042, 6.07865991507723936645028123105, 7.25568037855209186967890077724, 7.86559320817848651915440246046, 9.193686614882849819133480244343, 10.54803090851385578748877339536, 11.09532776200101792460887264893, 11.88873485670895070084127589754

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