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Properties

Label 2-325-65.8-c1-0-10
Degree $2$
Conductor $325$
Sign $0.678 - 0.734i$
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  + 1.68·2-s + (1.99 + 1.99i)3-s + 0.844·4-s + (3.36 + 3.36i)6-s − 1.68i·7-s − 1.94·8-s + 4.97i·9-s + (−1.38 + 1.38i)11-s + (1.68 + 1.68i)12-s + (3.53 − 0.723i)13-s − 2.84i·14-s − 4.97·16-s + (−4.40 − 4.40i)17-s + 8.39i·18-s + (5.89 − 5.89i)19-s + ⋯
L(s)  = 1  + 1.19·2-s + (1.15 + 1.15i)3-s + 0.422·4-s + (1.37 + 1.37i)6-s − 0.637i·7-s − 0.688·8-s + 1.65i·9-s + (−0.418 + 0.418i)11-s + (0.486 + 0.486i)12-s + (0.979 − 0.200i)13-s − 0.760i·14-s − 1.24·16-s + (−1.06 − 1.06i)17-s + 1.97i·18-s + (1.35 − 1.35i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69000 + 1.17670i\)
\(L(\frac12)\) \(\approx\) \(2.69000 + 1.17670i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.53 + 0.723i)T \)
good2 \( 1 - 1.68T + 2T^{2} \)
3 \( 1 + (-1.99 - 1.99i)T + 3iT^{2} \)
7 \( 1 + 1.68iT - 7T^{2} \)
11 \( 1 + (1.38 - 1.38i)T - 11iT^{2} \)
17 \( 1 + (4.40 + 4.40i)T + 17iT^{2} \)
19 \( 1 + (-5.89 + 5.89i)T - 19iT^{2} \)
23 \( 1 + (3.80 - 3.80i)T - 23iT^{2} \)
29 \( 1 - 1.60iT - 29T^{2} \)
31 \( 1 + (1.22 + 1.22i)T + 31iT^{2} \)
37 \( 1 - 7.86iT - 37T^{2} \)
41 \( 1 + (-1.76 - 1.76i)T + 41iT^{2} \)
43 \( 1 + (0.452 - 0.452i)T - 43iT^{2} \)
47 \( 1 + 0.422iT - 47T^{2} \)
53 \( 1 + (9.85 + 9.85i)T + 53iT^{2} \)
59 \( 1 + (0.149 + 0.149i)T + 59iT^{2} \)
61 \( 1 + 3.87T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 + (-6.25 - 6.25i)T + 71iT^{2} \)
73 \( 1 - 9.09T + 73T^{2} \)
79 \( 1 - 4.71iT - 79T^{2} \)
83 \( 1 - 4.89iT - 83T^{2} \)
89 \( 1 + (-4.59 - 4.59i)T + 89iT^{2} \)
97 \( 1 + 1.47T + 97T^{2} \)
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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

−11.69667181455266754493700025370, −10.89089243439791080120134583144, −9.669227499311964301631939319872, −9.170721362003247455391086665655, −8.031007052893957722421590524203, −6.80960846107219618628338236047, −5.23252905611559750879801802779, −4.52007034838778371837222782212, −3.57518226385474190808134522596, −2.73509979455922683449105648318, 1.92973748290490875195319239585, 3.10402539865077773896401950877, 4.05705658484394687176962393486, 5.76183327143822815611692234308, 6.33865519608341952735547872622, 7.71753552628530059015087339989, 8.547610733245031175000215903558, 9.223617117626711330761476723061, 10.84684563117615683669376167430, 12.13568562308511979442307488067

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