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Properties

Label 2-325-13.4-c1-0-17
Degree $2$
Conductor $325$
Sign $-0.0308 + 0.999i$
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  + (0.576 + 0.333i)2-s + (1.24 − 2.15i)3-s + (−0.778 − 1.34i)4-s + (1.43 − 0.826i)6-s + (0.509 − 0.293i)7-s − 2.36i·8-s + (−1.58 − 2.74i)9-s + (−2.58 − 1.49i)11-s − 3.86·12-s + (−1.93 + 3.04i)13-s + 0.391·14-s + (−0.767 + 1.32i)16-s + (3.28 + 5.69i)17-s − 2.10i·18-s + (5.19 − 3.00i)19-s + ⋯
L(s)  = 1  + (0.407 + 0.235i)2-s + (0.716 − 1.24i)3-s + (−0.389 − 0.673i)4-s + (0.584 − 0.337i)6-s + (0.192 − 0.111i)7-s − 0.837i·8-s + (−0.527 − 0.913i)9-s + (−0.779 − 0.450i)11-s − 1.11·12-s + (−0.537 + 0.843i)13-s + 0.104·14-s + (−0.191 + 0.332i)16-s + (0.797 + 1.38i)17-s − 0.496i·18-s + (1.19 − 0.688i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0308 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0308 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.0308 + 0.999i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.0308 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23855 - 1.27740i\)
\(L(\frac12)\) \(\approx\) \(1.23855 - 1.27740i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.93 - 3.04i)T \)
good2 \( 1 + (-0.576 - 0.333i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.24 + 2.15i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.509 + 0.293i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.58 + 1.49i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.28 - 5.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.19 + 3.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.335 + 0.580i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.94 + 6.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.53iT - 31T^{2} \)
37 \( 1 + (-6.70 - 3.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.629 + 0.363i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.503 + 0.871i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.15iT - 47T^{2} \)
53 \( 1 - 4.94T + 53T^{2} \)
59 \( 1 + (12.3 - 7.13i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.03 - 6.98i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.551 + 0.318i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.79 - 4.50i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 16.8iT - 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 0.370iT - 83T^{2} \)
89 \( 1 + (8.00 + 4.61i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.99 - 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.60955578980813973137119610581, −10.33452454113281046062276924177, −9.429035527544014024326145279200, −8.297970030173016266872837168543, −7.55219030908060388058194221737, −6.51927950745228286596494842919, −5.57477379520356922215043140278, −4.27376828093807937627347183358, −2.67108408394628950730925545115, −1.18745379162823854317512857758, 2.84333527637704160232273294602, 3.39638628088924015855668501698, 4.85525295171190808436905588720, 5.20056777795459939875783911704, 7.47006987576592602232843836025, 8.102312959273602975912125810764, 9.235379358963058380318595613563, 9.863668068311630156028305482420, 10.79189254470658261506138842327, 12.00163354858605316057401591437

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