L-function 1-315-315.32-r0-0-0

• Label: 1-315-315.32-r0-0-0
• Degree: $1$
• Conductor: $315$
• Sign: $-0.927 + 0.373i$
• Analytic cond.: $1.46285$
• Root an. cond.: $1.46285$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.866 + 0.5i)2^-s + (0.5 − 0.866i)4^-s + i·8^-s − 11^-s + (−0.866 + 0.5i)13^-s + (−0.5 − 0.866i)16^-s + (−0.866 + 0.5i)17^-s + (0.5 − 0.866i)19^-s + (0.866 − 0.5i)22^-s + i·23^-s + (0.5 − 0.866i)26^-s + (−0.5 + 0.866i)29^-s + (−0.5 + 0.866i)31^-s + (0.866 + 0.5i)32^-s + (0.5 − 0.866i)34^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign:	$-0.927 + 0.373i$
Analytic conductor:	\(1.46285\)
Root analytic conductor:	\(1.46285\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{315} (32, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 315,\ (0:\ ),\ -0.927 + 0.373i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.06033475863 + 0.3116489427i\)
\(L(1)\)	\(\approx\)	\(0.5037290851 + 0.1793943484i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)

	$p$	$F_p(T)$
bad	3	\( 1 \)
	5	\( 1 \)
	7	\( 1 \)
good	2	\( 1 + (-0.866 + 0.5i)T \)
	11	\( 1 - T \)
	13	\( 1 + (-0.866 + 0.5i)T \)
	17	\( 1 + (-0.866 + 0.5i)T \)
	19	\( 1 + (0.5 - 0.866i)T \)
	23	\( 1 + iT \)
	29	\( 1 + (-0.5 + 0.866i)T \)
	31	\( 1 + (-0.5 + 0.866i)T \)
	37	\( 1 + (-0.866 - 0.5i)T \)

Imaginary part of the first few zeros on the critical line

−24.81691738065062368115689812960, −24.275936029477748031419381672265, −22.73364112668188304589451374778, −22.10707714451699911690118111036, −20.80146521230657006807077792551, −20.466636978219789761137303583389, −19.321079484019608874849568571825, −18.50284668490545082091836364485, −17.755145577554383265219048080127, −16.81000488195463795851920423067, −15.90790489729107130861798988259, −15.006300979025657748585620351429, −13.559639523652657174025914934910, −12.637798142052239600539916088161, −11.77436614905722972238423611366, −10.66357034637802194439654355325, −9.98914702087436462156537336315, −8.93462512086578721889785564499, −7.8751898528847714211297435305, −7.16117981799885950592499372294, −5.71868416335784632752684426396, −4.33740242732157043748293840454, −2.93808714559449582253573178251, −2.05886355607982253170862122243, −0.25358185061230807462006199457, 1.659650346242510939403172232, 2.85505485518029054316377912632, 4.71883406027589769644770074814, 5.63462391328415859303369079908, 6.936042610067118413013223931586, 7.59589540279136305637225713016, 8.78570696617478129410844593615, 9.57021770289690763585573596383, 10.6430437238123330787689389895, 11.414863167209161025871926015105, 12.748257794915709745461240329333, 13.88575117236056695131703124972, 14.92580025907964047609391069240, 15.73523067725194725599575044913, 16.52753921558350311810781734969, 17.66352247979827638579206653540, 18.102822691234226861206656879227, 19.38617433123705437703407217552, 19.82463093068715706718993023996, 21.03079560676504221244945360084, 21.973841050778924904645628769095, 23.234454765280117474076666567371, 24.07348681771668516842678895690, 24.61219919661122258060853231654, 25.938072604772538421517416372482

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