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Properties

Label 1-1205-1205.1024-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.998 - 0.0627i$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 6-s + (−0.866 + 0.5i)7-s + 8-s + (−0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (−0.5 − 0.866i)12-s + (−0.866 − 0.5i)13-s + (0.866 + 0.5i)14-s + (−0.5 − 0.866i)16-s + i·17-s + (−0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 6-s + (−0.866 + 0.5i)7-s + 8-s + (−0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (−0.5 − 0.866i)12-s + (−0.866 − 0.5i)13-s + (0.866 + 0.5i)14-s + (−0.5 − 0.866i)16-s + i·17-s + (−0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.998 - 0.0627i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.998 - 0.0627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0004991907496 + 0.01588330954i\)
\(L(\frac12)\) \(\approx\) \(0.0004991907496 + 0.01588330954i\)
\(L(1)\) \(\approx\) \(0.5156413535 - 0.03224328516i\)
\(L(1)\) \(\approx\) \(0.5156413535 - 0.03224328516i\)

Euler product

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

−22.10229473939045832666443265852, −20.36442108163552708909511173221, −19.73516726411360714684973725102, −19.0686640279359524765332787159, −18.48061774485969665071459778395, −17.402559716117592686766669397028, −17.16333784194904443327308582986, −16.30964661266197974611996121276, −15.66463662874993484339689341828, −14.44113359296857241597287088985, −13.89163396016327986075285681686, −13.19949767402214315560011183415, −12.11595326526986765471868960861, −11.537864609391868055908596227111, −10.2720468376215534231806806235, −9.63302109356675685186622983224, −8.90167387791263504023129644035, −7.49500769538866971229206633270, −7.289224760791430742404280693898, −6.54211765638312579331793135292, −5.66538788097397518040026451131, −4.83955139582561120616123509476, −3.68241613352014315977267480876, −2.17145574273220142537125695612, −1.16234714223302667523591861269, 0.00946029611312653835990747648, 1.340799376217902877966257487943, 2.86730419367504723092117594816, 3.3234326051875729141624851394, 4.33110975494047190980401964258, 5.24162347820257991634819850724, 6.24273091833420833027274966710, 7.18999839098350298384347883741, 8.61664311162227487909741881848, 9.01391035746879604201397181701, 9.90488896628929553954440092465, 10.46202598648391584942367454333, 11.26601014189659945409056974395, 12.307013622717625140235320656548, 12.4154354858808069559784463388, 13.715031797076914293670602973504, 14.678229615879844091507967926465, 15.55348446186435767675678874877, 16.49820052115102052605859399292, 16.87392584488117655148717946264, 17.75965172009343491962162496361, 18.498673698304920626624814849240, 19.55562743593438753591583329731, 19.869871475723031146241315278049, 20.77822876389544620607509541634

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