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Properties

Label 1-2415-2415.1244-r0-0-0
Degree $1$
Conductor $2415$
Sign $0.996 - 0.0861i$
Analytic cond. $11.2152$
Root an. cond. $11.2152$
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  + (0.580 − 0.814i)2-s + (−0.327 − 0.945i)4-s + (−0.959 − 0.281i)8-s + (−0.580 − 0.814i)11-s + (−0.142 + 0.989i)13-s + (−0.786 + 0.618i)16-s + (−0.981 + 0.189i)17-s + (−0.981 − 0.189i)19-s − 22-s + (0.723 + 0.690i)26-s + (0.654 + 0.755i)29-s + (−0.723 + 0.690i)31-s + (0.0475 + 0.998i)32-s + (−0.415 + 0.909i)34-s + (0.888 + 0.458i)37-s + (−0.723 + 0.690i)38-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s + (−0.327 − 0.945i)4-s + (−0.959 − 0.281i)8-s + (−0.580 − 0.814i)11-s + (−0.142 + 0.989i)13-s + (−0.786 + 0.618i)16-s + (−0.981 + 0.189i)17-s + (−0.981 − 0.189i)19-s − 22-s + (0.723 + 0.690i)26-s + (0.654 + 0.755i)29-s + (−0.723 + 0.690i)31-s + (0.0475 + 0.998i)32-s + (−0.415 + 0.909i)34-s + (0.888 + 0.458i)37-s + (−0.723 + 0.690i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.996 - 0.0861i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (1244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ 0.996 - 0.0861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.331534146 - 0.05746882752i\)
\(L(\frac12)\) \(\approx\) \(1.331534146 - 0.05746882752i\)
\(L(1)\) \(\approx\) \(1.039407885 - 0.4423627939i\)
\(L(1)\) \(\approx\) \(1.039407885 - 0.4423627939i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.580 - 0.814i)T \)
11 \( 1 + (-0.580 - 0.814i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (-0.981 + 0.189i)T \)
19 \( 1 + (-0.981 - 0.189i)T \)
29 \( 1 + (0.654 + 0.755i)T \)
31 \( 1 + (-0.723 + 0.690i)T \)
37 \( 1 + (0.888 + 0.458i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.928 + 0.371i)T \)
59 \( 1 + (-0.786 - 0.618i)T \)
61 \( 1 + (-0.235 - 0.971i)T \)
67 \( 1 + (0.995 + 0.0950i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (-0.327 - 0.945i)T \)
79 \( 1 + (0.928 - 0.371i)T \)
83 \( 1 + (-0.841 + 0.540i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−19.68615379256860992993315680172, −18.566106424140664377622395011078, −17.86407041551667524147397138845, −17.42008344969325156669400347095, −16.62634630582685766274258381076, −15.73933728522840437506794576052, −15.270815718857448091276391611930, −14.70939656344782468707828611915, −13.81941800884896603423676236692, −12.899269328810006051623301414900, −12.7826778925274420954006144290, −11.758015871704404093874107343757, −10.85573059227460934025993060433, −10.0313901510736541045581100251, −9.1019753520010765906583386835, −8.34029605228848769028638431552, −7.58345986910918129360676762926, −7.0197221538372730172452301392, −6.01671013145004516760755597559, −5.480227744641327272785972333091, −4.442796359009432140706942710395, −4.05300556502956794970824402811, −2.7308186116114509142566793772, −2.236683501339067834776248792784, −0.40131579402749054131933981949, 0.93647012852739437427581944039, 2.04210286613764515641443715648, 2.6685564959874735377679761749, 3.63664614666313601245506018043, 4.44867241001488591830346157079, 5.05009750911696114576436226932, 6.1494592322791257837001641483, 6.569347003906484433141265705736, 7.77812872473083291830310354579, 8.90744412575144454848728389965, 9.15975988403427410598299075426, 10.37997414688811336173955949615, 10.90368617256339179615587263227, 11.43423311497833872171015586141, 12.43167128647286235051559665595, 12.93046802074913558844674733200, 13.755753489191242375719483865249, 14.27056105039759186132231417737, 15.088101774569648200501282095316, 15.84971096174777835762469812380, 16.56331028383288202819575416553, 17.57555321138731576657414561002, 18.30004451866360947618545404585, 18.996089949974757082618990738, 19.55547071314099441024108623472

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