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Properties

Label 2-775-5.4-c1-0-13
Degree $2$
Conductor $775$
Sign $-0.447 - 0.894i$
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  + 2i·2-s − i·3-s − 2·4-s + 2·6-s + 2i·7-s + 2·9-s + 2·11-s + 2i·12-s − 6i·13-s − 4·14-s − 4·16-s + 7i·17-s + 4i·18-s + 5·19-s + 2·21-s + 4i·22-s + ⋯
L(s)  = 1  + 1.41i·2-s − 0.577i·3-s − 4-s + 0.816·6-s + 0.755i·7-s + 0.666·9-s + 0.603·11-s + 0.577i·12-s − 1.66i·13-s − 1.06·14-s − 16-s + 1.69i·17-s + 0.942i·18-s + 1.14·19-s + 0.436·21-s + 0.852i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.865056 + 1.39969i\)
\(L(\frac12)\) \(\approx\) \(0.865056 + 1.39969i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 - 2iT - 2T^{2} \)
3 \( 1 + iT - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 9iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - 5T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 18iT - 97T^{2} \)
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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

−10.39819776369597752601159342084, −9.487120187023090282700677371811, −8.442173655573431288336976626063, −7.88379971403984484165128329441, −7.18334230696281159778892921627, −6.11211468712164572003204762363, −5.74558786990638540020116643697, −4.59842055452426993615804526995, −3.14580622908520450550634039847, −1.49819812464331558678378188320, 0.958300289032200049219196324017, 2.22230345840774583369564136537, 3.57856937666683524762731395444, 4.20313846453689961028208740922, 5.00937364099603027515078673824, 6.84703808377722316437417446864, 7.19589620735259790174873394243, 8.963485394908542850779906998827, 9.460421666197571723487895421419, 10.09322604211960826311946000607

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