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Properties

Label 2-775-155.123-c1-0-36
Degree $2$
Conductor $775$
Sign $-0.790 + 0.612i$
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  + (1.72 − 1.72i)2-s + (−2.01 + 2.01i)3-s − 3.95i·4-s + 6.93i·6-s + (−0.725 + 0.725i)7-s + (−3.37 − 3.37i)8-s − 5.08i·9-s − 4.60i·11-s + (7.95 + 7.95i)12-s + (−1.16 + 1.16i)13-s + 2.50i·14-s − 3.74·16-s + (−3.76 − 3.76i)17-s + (−8.77 − 8.77i)18-s − 3.87i·19-s + ⋯
L(s)  = 1  + (1.22 − 1.22i)2-s + (−1.16 + 1.16i)3-s − 1.97i·4-s + 2.83i·6-s + (−0.274 + 0.274i)7-s + (−1.19 − 1.19i)8-s − 1.69i·9-s − 1.38i·11-s + (2.29 + 2.29i)12-s + (−0.323 + 0.323i)13-s + 0.669i·14-s − 0.936·16-s + (−0.912 − 0.912i)17-s + (−2.06 − 2.06i)18-s − 0.888i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.790 + 0.612i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ -0.790 + 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461428 - 1.34818i\)
\(L(\frac12)\) \(\approx\) \(0.461428 - 1.34818i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.53 - 0.588i)T \)
good2 \( 1 + (-1.72 + 1.72i)T - 2iT^{2} \)
3 \( 1 + (2.01 - 2.01i)T - 3iT^{2} \)
7 \( 1 + (0.725 - 0.725i)T - 7iT^{2} \)
11 \( 1 + 4.60iT - 11T^{2} \)
13 \( 1 + (1.16 - 1.16i)T - 13iT^{2} \)
17 \( 1 + (3.76 + 3.76i)T + 17iT^{2} \)
19 \( 1 + 3.87iT - 19T^{2} \)
23 \( 1 + (-5.77 + 5.77i)T - 23iT^{2} \)
29 \( 1 - 4.95T + 29T^{2} \)
37 \( 1 + (4.78 + 4.78i)T + 37iT^{2} \)
41 \( 1 + 3.44T + 41T^{2} \)
43 \( 1 + (2.15 - 2.15i)T - 43iT^{2} \)
47 \( 1 + (1.45 - 1.45i)T - 47iT^{2} \)
53 \( 1 + (3.61 - 3.61i)T - 53iT^{2} \)
59 \( 1 + 4.09iT - 59T^{2} \)
61 \( 1 - 8.38iT - 61T^{2} \)
67 \( 1 + (-7.01 + 7.01i)T - 67iT^{2} \)
71 \( 1 - 1.76T + 71T^{2} \)
73 \( 1 + (1.92 - 1.92i)T - 73iT^{2} \)
79 \( 1 - 2.57T + 79T^{2} \)
83 \( 1 + (-1.48 + 1.48i)T - 83iT^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 + (-4.21 + 4.21i)T - 97iT^{2} \)
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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

−10.56753913511970975985547128863, −9.459452418494332015139080402731, −8.847565062074216363542976786160, −6.75436155346191937152371672693, −5.98032209453777258869504576453, −5.01131564348267371432198326452, −4.66615896167522996520367175450, −3.51882154693927235120081750628, −2.64718216696052170010239602105, −0.55018195508945231827894783702, 1.78947777798892599928802488631, 3.59722554719558451645270045594, 4.82475905344627039625521514065, 5.39334586428436923860496114442, 6.39143181583565992892761856868, 6.92332231329647475379927049512, 7.44873494427393139959942614475, 8.366443592908467994676509511549, 9.914232046811880893606112113072, 10.92992989051053711699657347012

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