Properties

Label 2-1925-385.208-c0-0-1
Degree $2$
Conductor $1925$
Sign $0.772 - 0.635i$
Analytic cond. $0.960700$
Root an. cond. $0.980153$
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  + (−0.965 + 0.258i)2-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.5 − 0.866i)11-s + (1.22 + 1.22i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (0.965 + 0.258i)18-s + (0.707 + 0.707i)22-s + (−1.49 − 0.866i)26-s + (1.5 − 0.866i)31-s + (0.707 − 0.707i)43-s + 1.00i·49-s + 0.999·56-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.5 − 0.866i)11-s + (1.22 + 1.22i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (0.965 + 0.258i)18-s + (0.707 + 0.707i)22-s + (−1.49 − 0.866i)26-s + (1.5 − 0.866i)31-s + (0.707 − 0.707i)43-s + 1.00i·49-s + 0.999·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.772 - 0.635i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.772 - 0.635i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6663987776\)
\(L(\frac12)\) \(\approx\) \(0.6663987776\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
3 \( 1 + (0.866 + 0.5i)T^{2} \)
13 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
89 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.111409934785891487394802365445, −8.620345160379340901774229889217, −8.327560554515529022719506606111, −7.35183324109194090073564341983, −6.28317961894053315161225176589, −5.76066758841463795084346311750, −4.55761231543734524211270110022, −3.65844659407670652915571274315, −2.43833824575708422277377111812, −1.05749885022927163648463980764, 0.916510227195385905098943437994, 2.06765426917651471957245856196, 3.23349741532375650074172015590, 4.57978537725006966987753100316, 5.14378359628300897550399763439, 6.11653805523956191504989034023, 7.35821893994503716277185259360, 8.103570559006781653853434353207, 8.318362845604654385132785449155, 9.271432042953791997888296824423

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