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Properties

Label 2-775-155.69-c1-0-19
Degree $2$
Conductor $775$
Sign $0.607 - 0.794i$
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  + (1.75 − 0.571i)2-s + (0.103 + 0.488i)3-s + (1.15 − 0.836i)4-s + (0.461 + 0.800i)6-s + (−1.51 + 3.41i)7-s + (−0.627 + 0.863i)8-s + (2.51 − 1.11i)9-s + (0.194 + 1.84i)11-s + (0.528 + 0.475i)12-s + (−3.85 + 3.46i)13-s + (−0.721 + 6.86i)14-s + (−1.48 + 4.58i)16-s + (5.63 + 0.592i)17-s + (3.78 − 3.40i)18-s + (0.962 − 1.06i)19-s + ⋯
L(s)  = 1  + (1.24 − 0.404i)2-s + (0.0599 + 0.282i)3-s + (0.575 − 0.418i)4-s + (0.188 + 0.326i)6-s + (−0.573 + 1.28i)7-s + (−0.221 + 0.305i)8-s + (0.837 − 0.372i)9-s + (0.0585 + 0.556i)11-s + (0.152 + 0.137i)12-s + (−1.06 + 0.962i)13-s + (−0.192 + 1.83i)14-s + (−0.372 + 1.14i)16-s + (1.36 + 0.143i)17-s + (0.891 − 0.802i)18-s + (0.220 − 0.245i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.607 - 0.794i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.607 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39677 + 1.18521i\)
\(L(\frac12)\) \(\approx\) \(2.39677 + 1.18521i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 + 5.26i)T \)
good2 \( 1 + (-1.75 + 0.571i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.103 - 0.488i)T + (-2.74 + 1.22i)T^{2} \)
7 \( 1 + (1.51 - 3.41i)T + (-4.68 - 5.20i)T^{2} \)
11 \( 1 + (-0.194 - 1.84i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (3.85 - 3.46i)T + (1.35 - 12.9i)T^{2} \)
17 \( 1 + (-5.63 - 0.592i)T + (16.6 + 3.53i)T^{2} \)
19 \( 1 + (-0.962 + 1.06i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (-2.08 + 2.86i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.424 + 1.30i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (-3.89 + 2.25i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.61 - 0.981i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (4.87 + 4.38i)T + (4.49 + 42.7i)T^{2} \)
47 \( 1 + (4.02 + 1.30i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.27 - 11.8i)T + (-35.4 + 39.3i)T^{2} \)
59 \( 1 + (2.13 - 0.453i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + 2.68T + 61T^{2} \)
67 \( 1 + (2.49 + 1.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.22 + 3.66i)T + (47.5 - 52.7i)T^{2} \)
73 \( 1 + (-4.18 + 0.439i)T + (71.4 - 15.1i)T^{2} \)
79 \( 1 + (-1.17 + 11.1i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (-2.17 + 10.2i)T + (-75.8 - 33.7i)T^{2} \)
89 \( 1 + (2.18 - 1.58i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.87 - 6.70i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.48921887036193574508020436461, −9.460443971336793355893464746189, −9.166626886175698463175420105614, −7.72400191841233644400552264510, −6.65366869148958145432822210651, −5.75227160917413433265303644556, −4.87812696738207111236467986284, −4.08503981521866023046555359535, −3.00398526880681452647395384627, −2.07700487335614576355335197671, 0.949193089456301467256426790695, 3.05584856196172052479441235960, 3.73832637025215067751766566583, 4.85384079814797422829132261390, 5.56118955654795315757826355107, 6.73228055880659578018593121591, 7.31009302819486942211901761884, 8.011765574779440493216325024993, 9.781124478039023558951467705236, 9.975766830125968223658166785375

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