Properties

Degree 1
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.850 + 0.525i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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 + ⋯
L(s,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.850 + 0.525i$
motivic weight  =  \(0\)
character  :  $\chi_{105} (62, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 105,\ (1:\ ),\ 0.850 + 0.525i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9751193533 + 0.2770109735i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9751193533 + 0.2770109735i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8439644980 - 0.1992329921i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8439644980 - 0.1992329921i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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