L-function 1-105-105.83-r1-0-0

• Label: 1-105-105.83-r1-0-0
• Degree: $1$
• Conductor: $105$
• Sign: $0.850 - 0.525i$
• Analytic cond.: $11.2838$
• Root an. cond.: $11.2838$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign:	$0.850 - 0.525i$
Analytic conductor:	\(11.2838\)
Root analytic conductor:	\(11.2838\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{105} (83, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 105,\ (1:\ ),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.9751193533 - 0.2770109735i\)
\(L(1)\)	\(\approx\)	\(0.8439644980 + 0.1992329921i\)

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	\( 1 \)
	11	\( 1 - T \)
	13	\( 1 \)
	17	\( 1 \)
	19	\( 1 \)
	23	\( 1 - iT \)
	29	\( 1 \)
	31	\( 1 \)
	37	\( 1 - T \)

Imaginary part of the first few zeros on the critical line

−29.281772712175060894556577932018, −28.7803542822713027662624067000, −27.68915123418808460429519022151, −26.59128790396878663945972776246, −25.797978175479164416069098744693, −24.02273806519508520735161875703, −23.31154208887711050229175138388, −21.965331944079479711939263051881, −21.25871821659880649466736421449, −20.18213979376223114596915264427, −19.137603037643431779506549092930, −18.23627020972632811165172509106, −17.141685661949906251734544241922, −15.72312168212251490904815690420, −14.26629951072653476900934713745, −13.30592804289734548550583790756, −12.19549982816182197562284435915, −11.11111527641253730828016001464, −10.04218328455248515303816725316, −8.9099309481894105865058220559, −7.605609188772651339637798025904, −5.69034467675670722687335627643, −4.37062071932560717998920505641, −2.98572939418016749054447739092, −1.510770454804617882590380067103, 0.46437325139824020569058668744, 3.030373442549094186896814012, 4.78051390801352578558561417740, 5.74966474735835216193008437911, 7.22906667923874713929723709758, 8.132284663556032389275852138934, 9.4569651423998353165289284769, 10.637739135653038899602409697422, 12.42616977845708052709915870269, 13.44905629494587222234060546734, 14.53358474920802562160617879169, 15.707585581133894715388922723407, 16.39866786565939171957754227745, 17.90369482256252119800790377661, 18.350680631649049034387552128219, 19.915245245985452392312379172127, 21.17315758624303658577726834124, 22.50139939491500554979277918599, 23.17182134834042099748864689168, 24.39462741152553208553202970676, 25.137368666182935004623860006396, 26.27061327884823891638222921579, 27.05416029370224432470192045493, 28.135794837945707317288197925927, 29.2765774250647467467420299895

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