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Properties

Label 2-325-65.64-c1-0-10
Degree $2$
Conductor $325$
Sign $0.124 + 0.992i$
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-s − 2i·3-s − 4-s + 2i·6-s + 5·7-s + 3·8-s − 9-s + 3i·11-s + 2i·12-s + (−2 − 3i)13-s − 5·14-s − 16-s − 5i·17-s + 18-s − 4i·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15i·3-s − 0.5·4-s + 0.816i·6-s + 1.88·7-s + 1.06·8-s − 0.333·9-s + 0.904i·11-s + 0.577i·12-s + (−0.554 − 0.832i)13-s − 1.33·14-s − 0.250·16-s − 1.21i·17-s + 0.235·18-s − 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.124 + 0.992i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.124 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.716267 - 0.632308i\)
\(L(\frac12)\) \(\approx\) \(0.716267 - 0.632308i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 3i)T \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 - 5T + 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + iT - 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 7T + 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 - 3iT - 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + 8iT - 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 - 18iT - 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−11.46734096261164614453853057994, −10.42012512599753505814830928101, −9.391692171468682841177486098011, −8.350847253418280992659148680446, −7.55906263091254102720028666402, −7.20677905564976580363652819523, −5.24727016759936878409990803024, −4.57814842204522320964446182267, −2.20372613703075722135017250404, −1.03497245506912373904505206560, 1.64390557452230311934047307962, 3.98537530070758719262941598905, 4.60124737174930514912143557880, 5.59811001088471801996347295521, 7.44142842120530832435795620294, 8.498160332307600170901938960037, 8.803311490432469805165097172784, 10.09834917257210647979140745822, 10.63364356860836270969075467070, 11.38502158068163916495467263841

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