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Properties

Label 1-1205-1205.514-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.897 - 0.441i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s − 6-s + (−0.965 + 0.258i)7-s − i·8-s + (0.5 − 0.866i)9-s + (0.965 − 0.258i)11-s + (0.866 + 0.5i)12-s + (−0.965 − 0.258i)13-s + (0.965 + 0.258i)14-s + (−0.5 + 0.866i)16-s + (0.707 − 0.707i)17-s + (−0.866 + 0.5i)18-s + (−0.965 + 0.258i)19-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s − 6-s + (−0.965 + 0.258i)7-s − i·8-s + (0.5 − 0.866i)9-s + (0.965 − 0.258i)11-s + (0.866 + 0.5i)12-s + (−0.965 − 0.258i)13-s + (0.965 + 0.258i)14-s + (−0.5 + 0.866i)16-s + (0.707 − 0.707i)17-s + (−0.866 + 0.5i)18-s + (−0.965 + 0.258i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.897 - 0.441i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.897 - 0.441i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1931331670 - 0.8301219670i\)
\(L(\frac12)\) \(\approx\) \(0.1931331670 - 0.8301219670i\)
\(L(1)\) \(\approx\) \(0.7004703663 - 0.3965155660i\)
\(L(1)\) \(\approx\) \(0.7004703663 - 0.3965155660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
11 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (-0.965 - 0.258i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + iT \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.258 - 0.965i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.258 - 0.965i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−21.4155444996812679558671398826, −20.402695418084466318755876358612, −19.76711136521440786001343111862, −19.23156358513830646246657921497, −18.77170318889318988229340227638, −17.39926270140790614612924555866, −16.780212438153526547046064872972, −16.30850738610140828032162979670, −15.26329759442416371210641233579, −14.704448126395729428706288450275, −14.174676342530440768142945570045, −12.99869177950900798284451829519, −12.18992582335533407500166558079, −10.84554292738034407027514507559, −10.24876408671111629006680448415, −9.47849745793087060077059631322, −8.9709106305341880468511105007, −8.14677290993608102373271247386, −7.04595856140768779826985688809, −6.70874166280056487328240523315, −5.4225790029638811632242412632, −4.38609659920504430992168472101, −3.39952291399012761874670991517, −2.38278335111536981392939969373, −1.38480236166699313463129541999, 0.39962549820202353697423453458, 1.64935471157597664365796300593, 2.53005241101190343649636011571, 3.33099791748213809181825309460, 4.01636233891455472398800466446, 5.73583349231199665231945563323, 6.85777584278899042372468976045, 7.267484899783475111796150478768, 8.28349580546005201166927424974, 9.0993199170364403877718165059, 9.59517393064535001437268364750, 10.29489268450159928706887117724, 11.65208206701400993947409550914, 12.15394515580493030640247797703, 12.97407484054675673813267422124, 13.61586578808681107777132151245, 14.82959522531124198054047000781, 15.33322913197418488198727574092, 16.55984999493041481729768612490, 16.98708828069361202446420717145, 17.99174611530712743746555442579, 18.90357386640269844685696928121, 19.27000515701447827429152530780, 19.77361182065835906057592378571, 20.66197179517360742107971614569

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