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Properties

Label 2-775-31.25-c1-0-38
Degree $2$
Conductor $775$
Sign $-0.974 - 0.226i$
Analytic cond. $6.18840$
Root an. cond. $2.48765$
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  − 0.784·2-s + (−0.892 − 1.54i)3-s − 1.38·4-s + (0.699 + 1.21i)6-s + (−2.44 − 4.23i)7-s + 2.65·8-s + (−0.0921 + 0.159i)9-s + (2.31 − 4.01i)11-s + (1.23 + 2.13i)12-s + (2.21 − 3.83i)13-s + (1.92 + 3.32i)14-s + 0.686·16-s + (2.83 + 4.91i)17-s + (0.0723 − 0.125i)18-s + (−2.64 − 4.58i)19-s + ⋯
L(s)  = 1  − 0.554·2-s + (−0.515 − 0.892i)3-s − 0.692·4-s + (0.285 + 0.494i)6-s + (−0.925 − 1.60i)7-s + 0.938·8-s + (−0.0307 + 0.0532i)9-s + (0.699 − 1.21i)11-s + (0.356 + 0.617i)12-s + (0.613 − 1.06i)13-s + (0.513 + 0.888i)14-s + 0.171·16-s + (0.687 + 1.19i)17-s + (0.0170 − 0.0295i)18-s + (−0.607 − 1.05i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.974 - 0.226i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ -0.974 - 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0711285 + 0.620515i\)
\(L(\frac12)\) \(\approx\) \(0.0711285 + 0.620515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 + (-0.587 + 5.53i)T \)
good2 \( 1 + 0.784T + 2T^{2} \)
3 \( 1 + (0.892 + 1.54i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.44 + 4.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.31 + 4.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.21 + 3.83i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.83 - 4.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.64 + 4.58i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.98T + 23T^{2} \)
29 \( 1 + 0.0152T + 29T^{2} \)
37 \( 1 + (2.48 + 4.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.36 - 2.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.811 - 1.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.40T + 47T^{2} \)
53 \( 1 + (-1.04 + 1.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.59 - 6.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.59T + 61T^{2} \)
67 \( 1 + (-6.13 + 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.37 - 4.11i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.03 - 8.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.55 - 6.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.10 - 8.84i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.934T + 89T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−9.928292591215892016937757547530, −8.955000924933260706208427653433, −8.086304561003359081976518773753, −7.29586227217249733470762180452, −6.44260423124859088738982570428, −5.72075202773501933232614202550, −4.09850578155320244668903581259, −3.47456278165920949087007896074, −1.08393139127246802378367030309, −0.53502540407885654549354427538, 1.86092482013770743748949288033, 3.52863165992123393819599155062, 4.56039705912146277430099860617, 5.29578582017546549774336486577, 6.29389352726823940235814614691, 7.34805860113963831077342372421, 8.736488386449671346810290035444, 9.131745805161106053175508741169, 9.849066278384567618516712699711, 10.32838141088150690326837306191

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