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Properties

Label 1-2340-2340.527-r0-0-0
Degree $1$
Conductor $2340$
Sign $-0.957 + 0.289i$
Analytic cond. $10.8669$
Root an. cond. $10.8669$
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)7-s − i·11-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)19-s + (−0.866 − 0.5i)23-s + 29-s + (−0.866 + 0.5i)31-s + (−0.5 − 0.866i)37-s + (0.866 + 0.5i)41-s + (−0.866 + 0.5i)43-s + (−0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s + i·53-s − i·59-s + (−0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s − i·11-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)19-s + (−0.866 − 0.5i)23-s + 29-s + (−0.866 + 0.5i)31-s + (−0.5 − 0.866i)37-s + (0.866 + 0.5i)41-s + (−0.866 + 0.5i)43-s + (−0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s + i·53-s − i·59-s + (−0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(10.8669\)
Root analytic conductor: \(10.8669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2340,\ (0:\ ),\ -0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03400308228 + 0.2302008215i\)
\(L(\frac12)\) \(\approx\) \(0.03400308228 + 0.2302008215i\)
\(L(1)\) \(\approx\) \(0.8530604197 + 0.04332668654i\)
\(L(1)\) \(\approx\) \(0.8530604197 + 0.04332668654i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 - iT \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + iT \)
59 \( 1 - iT \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 + 0.866i)T \)
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

−19.47701009074319400597897221115, −18.44777512964025710940853359084, −17.68624256562132726378339821201, −17.29370962011282974843734488345, −16.50295142511984739137571496208, −15.62601841019439492205175421688, −14.90415140262869514342120801913, −14.316584004535535827849789791374, −13.44955661982526695900125365469, −12.85759747530781368805129630854, −11.97464243507188178375398883275, −11.27690643093990971568346645152, −10.29585330757945945846678969646, −10.07226502606085402951485433452, −8.864475856861167323572700161021, −8.18383293140880460149297473467, −7.373152552692229447484910316360, −6.72322976504866448866790266641, −5.85300018305152865840873762206, −4.73315361342978630342916055375, −4.25823344164557191317807460594, −3.42297183263503150930749449490, −2.050691070641875160304312171830, −1.61835349704281242284872101978, −0.0708658301994865053748660564, 1.33775989847718778988365612778, 2.37826308247612347570416782770, 2.97387191396700254112802630411, 4.19396396040170790062997403436, 4.85532830838403374481575455494, 5.82826184188164887465601774015, 6.364709827796452044473280570467, 7.35965619526502085181909019037, 8.39354097211580572350210583689, 8.72453673977806042804938767377, 9.52890332568331024298796670297, 10.68048616943492843504486184276, 11.127897352351896873993301953950, 11.9326475290155039053577290061, 12.64035320115744010665814601167, 13.46205479171650208279478877997, 14.24550560985055748403033292986, 14.82008307844077914661926726124, 15.79585064262530761081288545925, 16.12668480532680870842131430706, 17.12348102710548427613085131073, 18.00389721397489050547959960246, 18.28595155400395976544964811250, 19.30955866786884943424090847001, 19.75240513093995162123229939967

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