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Properties

Label 1-2015-2015.1487-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.661 + 0.749i$
Analytic cond. $216.541$
Root an. cond. $216.541$
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  + 2-s + i·3-s + 4-s + i·6-s − 7-s + 8-s − 9-s − i·11-s + i·12-s − 14-s + 16-s + i·17-s − 18-s + i·19-s − i·21-s − i·22-s + ⋯
L(s)  = 1  + 2-s + i·3-s + 4-s + i·6-s − 7-s + 8-s − 9-s − i·11-s + i·12-s − 14-s + 16-s + i·17-s − 18-s + i·19-s − i·21-s − i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.661 + 0.749i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.661 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.596597117 + 1.622509452i\)
\(L(\frac12)\) \(\approx\) \(3.596597117 + 1.622509452i\)
\(L(1)\) \(\approx\) \(1.753514531 + 0.5058380598i\)
\(L(1)\) \(\approx\) \(1.753514531 + 0.5058380598i\)

Euler product

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

−19.84676643829627981212433154830, −19.203888557571133180489829517753, −18.14551268153776827300629035784, −17.56868600269455912678070961750, −16.585367003137154928501354593805, −15.93940473638187285475220021028, −15.139364921601767308572193587490, −14.412098523331475931747431058947, −13.466397584856156718274721403013, −13.189509055855950424402609494688, −12.44948464740862230771364441360, −11.74807321815054259893472760714, −11.177938867922803855784720867771, −9.97022069921684312051058499475, −9.32212036650212203863433798970, −8.05608139603175839111221547091, −7.245806372797300667744353329206, −6.7688157287087545989426593217, −6.09497540994766769143331464996, −5.150123016289255374143108300331, −4.3839195512529349067502066856, −3.09225850813237301146394674688, −2.73413485099367771734590524675, −1.702284354086697993277905518651, −0.663591549586911215850138239604, 0.6788517236217621073868228316, 2.19128879994291499140778194344, 3.1307622937373064676460366102, 3.64947211343215928206719836865, 4.36033645415594915404458117263, 5.34695852051042868128376145395, 6.13528619792044991925381026109, 6.47264587089200406749089410167, 7.88116351137962859124868873820, 8.602822295289938326360832382092, 9.59643239142015413236564342273, 10.59295716366428396538300065748, 10.67450372058474692803741302600, 11.911974589902970887304868132612, 12.44545488611986369390852176309, 13.36430678825718170593420769405, 14.06651949959657463418849381440, 14.713676219900748273265705377725, 15.47846978639181391642903586113, 16.15427252918247562877432064963, 16.58996959650939372108056176663, 17.23168980636221023523362131491, 18.68779191241712400623904104334, 19.362021164877851487132858475422, 20.0357091321282377227724098286

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