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Properties

Label 2-775-155.134-c1-0-30
Degree $2$
Conductor $775$
Sign $0.930 - 0.365i$
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  + (0.206 − 0.284i)2-s + (2.87 − 0.302i)3-s + (0.579 + 1.78i)4-s + (0.508 − 0.880i)6-s + (−0.224 + 1.05i)7-s + (1.29 + 0.420i)8-s + (5.25 − 1.11i)9-s + (−1.62 + 1.80i)11-s + (2.20 + 4.95i)12-s + (1.16 − 2.62i)13-s + (0.254 + 0.282i)14-s + (−2.64 + 1.92i)16-s + (1.35 − 1.22i)17-s + (0.767 − 1.72i)18-s + (−1.93 + 0.861i)19-s + ⋯
L(s)  = 1  + (0.146 − 0.201i)2-s + (1.66 − 0.174i)3-s + (0.289 + 0.892i)4-s + (0.207 − 0.359i)6-s + (−0.0850 + 0.400i)7-s + (0.458 + 0.148i)8-s + (1.75 − 0.372i)9-s + (−0.490 + 0.544i)11-s + (0.637 + 1.43i)12-s + (0.324 − 0.728i)13-s + (0.0680 + 0.0755i)14-s + (−0.662 + 0.481i)16-s + (0.328 − 0.296i)17-s + (0.180 − 0.406i)18-s + (−0.443 + 0.197i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.930 - 0.365i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.930 - 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.97795 + 0.563680i\)
\(L(\frac12)\) \(\approx\) \(2.97795 + 0.563680i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 + 5.44i)T \)
good2 \( 1 + (-0.206 + 0.284i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-2.87 + 0.302i)T + (2.93 - 0.623i)T^{2} \)
7 \( 1 + (0.224 - 1.05i)T + (-6.39 - 2.84i)T^{2} \)
11 \( 1 + (1.62 - 1.80i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-1.16 + 2.62i)T + (-8.69 - 9.66i)T^{2} \)
17 \( 1 + (-1.35 + 1.22i)T + (1.77 - 16.9i)T^{2} \)
19 \( 1 + (1.93 - 0.861i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-0.420 - 0.136i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.55 + 1.85i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (2.72 + 1.57i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.726 + 6.90i)T + (-40.1 - 8.52i)T^{2} \)
43 \( 1 + (-3.42 - 7.68i)T + (-28.7 + 31.9i)T^{2} \)
47 \( 1 + (-4.67 - 6.44i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.03 + 4.86i)T + (-48.4 + 21.5i)T^{2} \)
59 \( 1 + (1.25 + 11.9i)T + (-57.7 + 12.2i)T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 + (5.57 - 3.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.64 - 0.348i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (10.6 + 9.60i)T + (7.63 + 72.6i)T^{2} \)
79 \( 1 + (-2.30 - 2.56i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (-12.7 - 1.34i)T + (81.1 + 17.2i)T^{2} \)
89 \( 1 + (0.698 + 2.14i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (3.24 - 1.05i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.22320757344484918810600799726, −9.264111339854792673882452170990, −8.610872907421729713207679271227, −7.62787163202177020607386360264, −7.55537639114015615203261497402, −6.07592562684779063742126078523, −4.55026304210251785495766904097, −3.55599501891780943012131427252, −2.77581432454039484835289073370, −1.98713857870675438245316399757, 1.47293405928433429612533616611, 2.61590210430647606326043197323, 3.70423404338169424521697179035, 4.67058310150547146861706765942, 5.89495812677190059182779592078, 6.96288175801200700484540052034, 7.67111715449466188045381044308, 8.756968893157243535216314656238, 9.173925900460587744245380415193, 10.36072802551251323727508967146

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