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Properties

Label 2-775-31.7-c1-0-31
Degree $2$
Conductor $775$
Sign $0.272 + 0.962i$
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  + (−0.380 − 1.17i)2-s + (2.02 − 0.431i)3-s + (0.389 − 0.283i)4-s + (−1.27 − 2.21i)6-s + (3.47 + 1.54i)7-s + (−2.47 − 1.79i)8-s + (1.19 − 0.531i)9-s + (0.393 + 3.74i)11-s + (0.668 − 0.742i)12-s + (1.76 + 1.95i)13-s + (0.489 − 4.66i)14-s + (−0.866 + 2.66i)16-s + (0.394 − 3.75i)17-s + (−1.07 − 1.19i)18-s + (4.08 − 4.53i)19-s + ⋯
L(s)  = 1  + (−0.269 − 0.828i)2-s + (1.17 − 0.249i)3-s + (0.194 − 0.141i)4-s + (−0.521 − 0.903i)6-s + (1.31 + 0.584i)7-s + (−0.874 − 0.635i)8-s + (0.397 − 0.177i)9-s + (0.118 + 1.12i)11-s + (0.193 − 0.214i)12-s + (0.489 + 0.543i)13-s + (0.130 − 1.24i)14-s + (−0.216 + 0.666i)16-s + (0.0956 − 0.910i)17-s + (−0.253 − 0.281i)18-s + (0.936 − 1.03i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.272 + 0.962i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.272 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92707 - 1.45655i\)
\(L(\frac12)\) \(\approx\) \(1.92707 - 1.45655i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.88 - 3.98i)T \)
good2 \( 1 + (0.380 + 1.17i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-2.02 + 0.431i)T + (2.74 - 1.22i)T^{2} \)
7 \( 1 + (-3.47 - 1.54i)T + (4.68 + 5.20i)T^{2} \)
11 \( 1 + (-0.393 - 3.74i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (-1.76 - 1.95i)T + (-1.35 + 12.9i)T^{2} \)
17 \( 1 + (-0.394 + 3.75i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (-4.08 + 4.53i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (-0.736 - 0.534i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.10 + 6.47i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (0.907 + 1.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.329 + 0.0700i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (2.59 - 2.88i)T + (-4.49 - 42.7i)T^{2} \)
47 \( 1 + (-0.367 + 1.13i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.14 + 0.953i)T + (35.4 - 39.3i)T^{2} \)
59 \( 1 + (7.60 - 1.61i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + 2.72T + 61T^{2} \)
67 \( 1 + (3.71 - 6.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.65 - 2.07i)T + (47.5 - 52.7i)T^{2} \)
73 \( 1 + (-0.563 - 5.36i)T + (-71.4 + 15.1i)T^{2} \)
79 \( 1 + (-1.01 + 9.68i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (8.21 + 1.74i)T + (75.8 + 33.7i)T^{2} \)
89 \( 1 + (4.12 - 2.99i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-8.82 + 6.41i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.974302316017232530245683662026, −9.214922634083252265927763913576, −8.753550211246302415156258445061, −7.62974461065608297465248701132, −7.01336783999271391312565704897, −5.56281911916442418432443616787, −4.49213136710224422976632021020, −3.12634982125920203799986567642, −2.24048370601894734413796709289, −1.51555686046893760107389963261, 1.61960789470263756630208142959, 3.14224075389555645231171859271, 3.80663279845126965875208173464, 5.34219924912877308427473076552, 6.13567091486493249448549112302, 7.48369718205490624580262071465, 7.969453428877856449317306244871, 8.504711775665484470762082866265, 9.177026860347860999298973556716, 10.49278483374839137071381600747

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