Properties

Label 2-605-121.16-c1-0-10
Degree $2$
Conductor $605$
Sign $0.0856 - 0.996i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (−1.42 − 0.163i)2-s + (1.67 + 1.21i)3-s + (0.0459 + 0.0106i)4-s + (−0.564 + 0.825i)5-s + (−2.18 − 2.00i)6-s + (2.67 + 2.75i)7-s + (2.63 + 0.938i)8-s + (0.400 + 1.23i)9-s + (0.936 − 1.08i)10-s + (2.07 + 2.58i)11-s + (0.0640 + 0.0739i)12-s + (0.816 − 1.35i)13-s + (−3.35 − 4.34i)14-s + (−1.95 + 0.696i)15-s + (−3.67 − 1.80i)16-s + (2.68 − 2.19i)17-s + ⋯
L(s)  = 1  + (−1.00 − 0.115i)2-s + (0.968 + 0.703i)3-s + (0.0229 + 0.00534i)4-s + (−0.252 + 0.369i)5-s + (−0.892 − 0.818i)6-s + (1.01 + 1.03i)7-s + (0.930 + 0.331i)8-s + (0.133 + 0.411i)9-s + (0.296 − 0.341i)10-s + (0.626 + 0.779i)11-s + (0.0185 + 0.0213i)12-s + (0.226 − 0.375i)13-s + (−0.895 − 1.16i)14-s + (−0.504 + 0.179i)15-s + (−0.918 − 0.451i)16-s + (0.652 − 0.533i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0856 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0856 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.0856 - 0.996i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.0856 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.873174 + 0.801293i\)
\(L(\frac12)\) \(\approx\) \(0.873174 + 0.801293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.564 - 0.825i)T \)
11 \( 1 + (-2.07 - 2.58i)T \)
good2 \( 1 + (1.42 + 0.163i)T + (1.94 + 0.452i)T^{2} \)
3 \( 1 + (-1.67 - 1.21i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-2.67 - 2.75i)T + (-0.199 + 6.99i)T^{2} \)
13 \( 1 + (-0.816 + 1.35i)T + (-6.06 - 11.4i)T^{2} \)
17 \( 1 + (-2.68 + 2.19i)T + (3.37 - 16.6i)T^{2} \)
19 \( 1 + (6.96 + 0.398i)T + (18.8 + 2.16i)T^{2} \)
23 \( 1 + (0.200 - 1.39i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + (1.34 - 3.44i)T + (-21.3 - 19.6i)T^{2} \)
31 \( 1 + (-8.55 - 3.61i)T + (21.6 + 22.2i)T^{2} \)
37 \( 1 + (0.827 - 9.63i)T + (-36.4 - 6.30i)T^{2} \)
41 \( 1 + (-5.51 + 3.11i)T + (21.1 - 35.1i)T^{2} \)
43 \( 1 + (12.3 - 3.61i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (-0.181 + 0.895i)T + (-43.2 - 18.2i)T^{2} \)
53 \( 1 + (-9.77 + 4.80i)T + (32.3 - 41.9i)T^{2} \)
59 \( 1 + (8.54 + 4.82i)T + (30.4 + 50.5i)T^{2} \)
61 \( 1 + (6.29 - 0.722i)T + (59.4 - 13.8i)T^{2} \)
67 \( 1 + (3.32 - 7.28i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-0.398 - 1.51i)T + (-61.8 + 34.9i)T^{2} \)
73 \( 1 + (-1.79 - 3.40i)T + (-41.2 + 60.2i)T^{2} \)
79 \( 1 + (0.242 + 8.50i)T + (-78.8 + 4.51i)T^{2} \)
83 \( 1 + (-0.580 - 0.100i)T + (78.1 + 27.8i)T^{2} \)
89 \( 1 + (-15.6 - 10.0i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (1.41 + 2.06i)T + (-35.1 + 90.3i)T^{2} \)
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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

−10.46132510916518294083310234328, −9.870668655605510801121267909998, −8.988572717329639538034447713236, −8.455383947699413738958974100167, −7.905101620278818331259843233653, −6.62108286045008365534123844695, −5.05693609571432114309262346035, −4.27302581000794902931824551896, −2.89938645446210390857737257112, −1.70263774778505831812741355894, 0.926696263785887521647635603848, 1.94182872165738250337618172180, 3.81698497286605035176222452475, 4.53492589809443393501966041066, 6.30190371351483002218625067489, 7.42429834874657606719011308083, 8.030651934513607459787083620846, 8.491051190384394132610943670921, 9.186062045138801663503772377152, 10.41473578232835200007356076592

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