Properties

Label 2-325-13.10-c1-0-17
Degree $2$
Conductor $325$
Sign $-0.770 - 0.637i$
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  + (0.769 − 0.443i)2-s + (−1.53 − 2.65i)3-s + (−0.605 + 1.04i)4-s + (−2.35 − 1.35i)6-s + (−3.08 − 1.78i)7-s + 2.85i·8-s + (−3.18 + 5.52i)9-s + (0.816 − 0.471i)11-s + 3.70·12-s + (−3.56 − 0.555i)13-s − 3.16·14-s + (0.0547 + 0.0947i)16-s + (−0.479 + 0.831i)17-s + 5.66i·18-s + (−2.89 − 1.66i)19-s + ⋯
L(s)  = 1  + (0.543 − 0.313i)2-s + (−0.883 − 1.53i)3-s + (−0.302 + 0.524i)4-s + (−0.961 − 0.555i)6-s + (−1.16 − 0.673i)7-s + 1.00i·8-s + (−1.06 + 1.84i)9-s + (0.246 − 0.142i)11-s + 1.07·12-s + (−0.988 − 0.154i)13-s − 0.845·14-s + (0.0136 + 0.0236i)16-s + (−0.116 + 0.201i)17-s + 1.33i·18-s + (−0.663 − 0.383i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.770 - 0.637i$
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: \((2,\ 325,\ (\ :1/2),\ -0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0953267 + 0.264776i\)
\(L(\frac12)\) \(\approx\) \(0.0953267 + 0.264776i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.56 + 0.555i)T \)
good2 \( 1 + (-0.769 + 0.443i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.53 + 2.65i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.08 + 1.78i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.816 + 0.471i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.479 - 0.831i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.89 + 1.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.29 - 5.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.19 + 7.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.13iT - 31T^{2} \)
37 \( 1 + (6.90 - 3.98i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.690 + 0.398i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.50 + 7.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + 7.44T + 53T^{2} \)
59 \( 1 + (-0.869 - 0.501i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.26 - 3.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.969 - 0.559i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.60 + 0.926i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.69iT - 73T^{2} \)
79 \( 1 - 5.64T + 79T^{2} \)
83 \( 1 - 0.187iT - 83T^{2} \)
89 \( 1 + (-8.21 + 4.74i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.3 - 8.27i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.54561757179187594022207129360, −10.42121352972408881145414155402, −9.113671241784779001444018193569, −7.76190146858437418675417894082, −7.14429517073381385582885915157, −6.19415495238089922542273918988, −5.11819462984925430631427033726, −3.66449334987301194101466478307, −2.25947251497395016891991727897, −0.17763735004191144525574948723, 3.22370093687291279056163528377, 4.42133047358289185705713128914, 5.13873845785512183463788664660, 6.06319154155066082360099347569, 6.79660716991968025513252094637, 9.070177206635803138697062739487, 9.414622486730879348549290729336, 10.31679370171620069719544716606, 10.98157980847168328759944066776, 12.41321975827335240063485092383

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