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Properties

Label 1-775-775.154-r1-0-0
Degree $1$
Conductor $775$
Sign $0.535 - 0.844i$
Analytic cond. $83.2853$
Root an. cond. $83.2853$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.809 + 0.587i)12-s + (−0.809 + 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s − 18-s + (0.309 − 0.951i)19-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.809 + 0.587i)12-s + (−0.809 + 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s − 18-s + (0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ 0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2001833674 + 0.1100517375i\)
\(L(\frac12)\) \(\approx\) \(-0.2001833674 + 0.1100517375i\)
\(L(1)\) \(\approx\) \(0.9169469716 + 0.9002758585i\)
\(L(1)\) \(\approx\) \(0.9169469716 + 0.9002758585i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 - T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.632687767311854513981116680020, −20.50430429807057595815806980560, −19.75577347406781807361636099829, −19.30852585864648285345182454461, −18.67389975079402631128531430252, −17.557860550748401741396656594803, −16.587134955177490855496470403373, −15.58521442737436065178596086359, −14.52681328185547949185584299049, −14.08982252807689760557343296522, −13.1353506526798149124816196356, −12.40632894975726341076893073209, −12.045203896754377291691352730574, −10.84729327079570175161924703839, −9.89556157825710906604084270100, −9.07919484514116990703399052658, −7.87122138772115032703807128598, −6.85883824911240048951458206388, −6.08044878709029291277828539419, −5.42875511463840038772166370777, −3.69217597445591735720352906900, −3.36837818180635924209101461492, −2.15707382709755524995295743474, −1.2224299198121420812337668125, −0.03469611555917954770008177150, 2.32381025476140423174563745480, 3.092695279217756800586287144560, 4.1145436681976118108977955668, 4.709337395161991985439158913033, 5.747113129146757898521495858175, 6.75670760765409961661906064692, 7.46133952144461934881837791243, 8.76930093858466818430167659554, 9.456077214908995979545866977518, 10.22111122547235368021286156840, 11.66132009753766148580288609528, 12.0488215649946125410064993285, 13.309036767102945805902599387816, 13.9810593163763652602660845724, 14.76112452160188860074614837811, 15.48426842019656995471804770881, 16.22683119820092133188730262677, 16.819726960141215702638717146048, 17.59034332562335610413250515065, 19.01094933522020032360888660523, 19.98455054154332823535680433421, 20.43570861261835302340018996738, 21.50899013218832787410552686571, 22.24494688094725088891713422026, 22.52719569758887956271278973028

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