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Properties

Label 2-925-185.43-c1-0-14
Degree $2$
Conductor $925$
Sign $0.148 - 0.988i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
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  − i·2-s + (−2 + 2i)3-s + 4-s + (2 + 2i)6-s − 3i·8-s − 5i·9-s + 4i·11-s + (−2 + 2i)12-s + 4i·13-s − 16-s + 2·17-s − 5·18-s + 4·22-s − 4i·23-s + (6 + 6i)24-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.15 + 1.15i)3-s + 0.5·4-s + (0.816 + 0.816i)6-s − 1.06i·8-s − 1.66i·9-s + 1.20i·11-s + (−0.577 + 0.577i)12-s + 1.10i·13-s − 0.250·16-s + 0.485·17-s − 1.17·18-s + 0.852·22-s − 0.834i·23-s + (1.22 + 1.22i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.148 - 0.988i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.148 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.780181 + 0.671637i\)
\(L(\frac12)\) \(\approx\) \(0.780181 + 0.671637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 + (-6 + i)T \)
good2 \( 1 + iT - 2T^{2} \)
3 \( 1 + (2 - 2i)T - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + (-1 - i)T + 29iT^{2} \)
31 \( 1 + (6 - 6i)T - 31iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + (8 - 8i)T - 47iT^{2} \)
53 \( 1 + (-9 - 9i)T + 53iT^{2} \)
59 \( 1 + (4 - 4i)T - 59iT^{2} \)
61 \( 1 + (-1 + i)T - 61iT^{2} \)
67 \( 1 + (6 + 6i)T + 67iT^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (11 - 11i)T - 73iT^{2} \)
79 \( 1 + (-6 + 6i)T - 79iT^{2} \)
83 \( 1 + (-2 - 2i)T + 83iT^{2} \)
89 \( 1 + (-1 - i)T + 89iT^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−10.44089392987637913027668278704, −9.702382855399671134386940558248, −9.171408600694710737707743373604, −7.53989996397920585443481198158, −6.65349216783918861382462476589, −5.91395047500636452854609912703, −4.65316115058200631171302220406, −4.22077069816108407412976478957, −2.91866971272006117984646547494, −1.47462677904579787382958312205, 0.57023921872925163619902988851, 1.98767477040767958338361305856, 3.37986431811293627670780505606, 5.38462908507685789230307794023, 5.62154317448142772340247265444, 6.40170516112092006994373073307, 7.26975271555471798583373292786, 7.83979133182160503979625978214, 8.616112214286251165235790292915, 10.14600525657120403437343948259

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