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Properties

Label 1-2415-2415.794-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: $\chi_{2415} (794, \cdot )$
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

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

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