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Properties

Label 1-1305-1305.389-r0-0-0
Degree $1$
Conductor $1305$
Sign $0.486 + 0.873i$
Analytic cond. $6.06039$
Root an. cond. $6.06039$
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 + (0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s − i·8-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + (0.866 + 0.5i)14-s + (−0.5 − 0.866i)16-s + i·17-s + i·19-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + i·26-s + 28-s + (−0.866 − 0.5i)31-s + (−0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s − i·8-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + (0.866 + 0.5i)14-s + (−0.5 − 0.866i)16-s + i·17-s + i·19-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + i·26-s + 28-s + (−0.866 − 0.5i)31-s + (−0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.486 + 0.873i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ 0.486 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.635742317 + 0.9616176399i\)
\(L(\frac12)\) \(\approx\) \(1.635742317 + 0.9616176399i\)
\(L(1)\) \(\approx\) \(1.509678019 + 0.02973512374i\)
\(L(1)\) \(\approx\) \(1.509678019 + 0.02973512374i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + iT \)
19 \( 1 + iT \)
23 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + iT \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−20.9419559333244680283904516200, −20.21390170748850300349773412229, −19.70602609548697278787560162845, −18.12509513114917665581314023932, −17.8943244374028454088378323676, −16.713036110142245999718690715896, −16.32087437158799592214665008097, −15.35372302300619754254623718904, −14.72227179908090963883753711241, −13.81605846025407058163780859416, −13.3575605436608663040592998471, −12.549623121760702437966310713908, −11.60123189177841955457067968718, −10.85693999802150822001839017100, −10.14594220393610726707272383812, −8.772770661511662659849956362070, −7.96874132294011221180956600150, −7.30251466140021569524310895940, −6.59849878544733612018338057419, −5.251502103977251022828901840146, −5.05711946400222932398019030548, −3.92054206191647180447458211143, −3.01401982641917728875857081743, −2.182271212022529619035517891803, −0.48959377929821538879779733464, 1.73301129651069257140107178073, 2.02976064609273122584144551379, 3.22774889265368116603061665894, 4.15732338968979873349938167183, 5.06220490536892468024765231253, 5.6698019947652648016190686823, 6.57149254394662350576319270356, 7.62428724140554569465734136717, 8.50100103256241983127947641616, 9.6934087144447244127983500574, 10.19049318362430214404459498769, 11.29893906713429622245305865019, 11.84053740216020474762252384002, 12.60356741384292487894803258267, 13.251369993041701015185700099805, 14.29162281277217278365793507398, 14.84587108326945635599707460068, 15.48011745203937593330383821214, 16.31627063190632206607358252085, 17.32537237070676283253048091115, 18.39044029316504114255458035642, 18.829798653594999946466295188455, 19.70121121999957423523154223772, 20.54666564644896868841650106589, 21.22215681401399801914615027854

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