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Properties

Label 2-775-31.25-c1-0-26
Degree $2$
Conductor $775$
Sign $-0.914 - 0.404i$
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  − 1.68·2-s + (−1.34 − 2.32i)3-s + 0.851·4-s + (2.27 + 3.93i)6-s + (2.52 + 4.37i)7-s + 1.93·8-s + (−2.11 + 3.66i)9-s + (−0.726 + 1.25i)11-s + (−1.14 − 1.98i)12-s + (−0.381 + 0.661i)13-s + (−4.26 − 7.38i)14-s − 4.97·16-s + (−2.68 − 4.65i)17-s + (3.57 − 6.18i)18-s + (0.756 + 1.31i)19-s + ⋯
L(s)  = 1  − 1.19·2-s + (−0.776 − 1.34i)3-s + 0.425·4-s + (0.926 + 1.60i)6-s + (0.954 + 1.65i)7-s + 0.685·8-s + (−0.704 + 1.22i)9-s + (−0.218 + 0.379i)11-s + (−0.330 − 0.572i)12-s + (−0.105 + 0.183i)13-s + (−1.14 − 1.97i)14-s − 1.24·16-s + (−0.651 − 1.12i)17-s + (0.841 − 1.45i)18-s + (0.173 + 0.300i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.914 - 0.404i$
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: \((2,\ 775,\ (\ :1/2),\ -0.914 - 0.404i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0100044 + 0.0473853i\)
\(L(\frac12)\) \(\approx\) \(0.0100044 + 0.0473853i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 5.32i)T \)
good2 \( 1 + 1.68T + 2T^{2} \)
3 \( 1 + (1.34 + 2.32i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.52 - 4.37i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.726 - 1.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.381 - 0.661i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.68 + 4.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.756 - 1.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.39T + 23T^{2} \)
29 \( 1 + 5.39T + 29T^{2} \)
37 \( 1 + (4.95 + 8.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.604 - 1.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.92 - 5.07i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.15T + 47T^{2} \)
53 \( 1 + (-0.783 + 1.35i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.78 + 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + (2.64 - 4.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.09 + 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.08 - 3.61i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.868 + 1.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.88 + 8.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 - 2.12T + 97T^{2} \)
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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

−9.508318989129672651583516354017, −8.992110998534309853539943084875, −7.889742909101097697979234535045, −7.62440648046927550246660670713, −6.48960790798164172759303489605, −5.56973348780578514927279804565, −4.73336083272291837004205451821, −2.28769307570449019944993929163, −1.69971701327008800451814256544, −0.04229658744805105931091864477, 1.44094480052751888157058356111, 3.75827816894242473562618442266, 4.45716615083197105396963816310, 5.22332188278719451608421896419, 6.59414151088953049182369587127, 7.63530036766795514905920417522, 8.348328011249800885973387261846, 9.255868238510070607612647421110, 10.22984709602109853385278081648, 10.58885099428871470625377898120

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