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Properties

Label 1-1309-1309.1165-r0-0-0
Degree $1$
Conductor $1309$
Sign $-0.574 - 0.818i$
Analytic cond. $6.07897$
Root an. cond. $6.07897$
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 + (−0.258 + 0.965i)3-s + (0.5 + 0.866i)4-s + (−0.965 + 0.258i)5-s + (0.707 − 0.707i)6-s − i·8-s + (−0.866 − 0.5i)9-s + (0.965 + 0.258i)10-s + (−0.965 + 0.258i)12-s − 13-s − i·15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)18-s + (0.866 + 0.5i)19-s + (−0.707 − 0.707i)20-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.258 + 0.965i)3-s + (0.5 + 0.866i)4-s + (−0.965 + 0.258i)5-s + (0.707 − 0.707i)6-s − i·8-s + (−0.866 − 0.5i)9-s + (0.965 + 0.258i)10-s + (−0.965 + 0.258i)12-s − 13-s − i·15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)18-s + (0.866 + 0.5i)19-s + (−0.707 − 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $-0.574 - 0.818i$
Analytic conductor: \(6.07897\)
Root analytic conductor: \(6.07897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (1165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (0:\ ),\ -0.574 - 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01703647950 + 0.03277686215i\)
\(L(\frac12)\) \(\approx\) \(0.01703647950 + 0.03277686215i\)
\(L(1)\) \(\approx\) \(0.4321563656 + 0.1144026982i\)
\(L(1)\) \(\approx\) \(0.4321563656 + 0.1144026982i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-0.965 + 0.258i)T \)
13 \( 1 - T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.258 + 0.965i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (-0.258 + 0.965i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.258 - 0.965i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.258 + 0.965i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−20.01442986771564134735011606395, −19.562843399595467678531239813944, −18.89974658970281718025477238983, −18.24470629483484552384116834439, −17.40003936982986251762606568066, −16.80431192098244490265345986938, −16.07998858886630770206821026235, −15.22135428609689339935930088606, −14.49484669022743626446650255781, −13.603743493566061061536343850594, −12.52727180046285166560676172095, −11.846706551698426377648508466979, −11.25388154945713321998991685605, −10.3089896466895626057375540705, −9.27043033719419556672450456116, −8.420723503177549700835536899200, −7.764991578605283041738239421108, −7.14547243600309282486129446251, −6.46844827387780758598574611644, −5.35223279193290119503197490775, −4.64928630699885800523277501623, −3.04027715578844008982608809191, −2.09971376042207953301697560832, −0.91745556713619494225347891598, −0.025989579440014545060591200217, 1.44842296634524926074070583693, 3.0392976569020307220701975740, 3.31664380762210318632485015598, 4.42653011104627670146020941054, 5.255485275790500731811950713177, 6.636207290362292614944783096, 7.42527958377984461512561659157, 8.23013516695473834333131360346, 9.08595030735874996732695539672, 9.83616457952488728968927762154, 10.55023447906737250796787033941, 11.22529030067215376567072392322, 12.03305732388922858141863111968, 12.39509167790623826028237036744, 13.923962804580358917577512874368, 14.83273574928289829648374227839, 15.6759851768989905808519282327, 16.07805755786693651546223271780, 16.95751077436381339965214889558, 17.58525945011150766760798801257, 18.4778753858829930245810528659, 19.315893569507068550202554007177, 19.99081556669166166249958346806, 20.442018300586542671314387445368, 21.4975850247546453064184828891

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