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Properties

Label 1-175-175.3-r0-0-0
Degree $1$
Conductor $175$
Sign $0.784 - 0.620i$
Analytic cond. $0.812696$
Root an. cond. $0.812696$
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  + (−0.406 − 0.913i)2-s + (−0.743 − 0.669i)3-s + (−0.669 + 0.743i)4-s + (−0.309 + 0.951i)6-s + (0.951 + 0.309i)8-s + (0.104 + 0.994i)9-s + (−0.104 + 0.994i)11-s + (0.994 − 0.104i)12-s + (0.587 + 0.809i)13-s + (−0.104 − 0.994i)16-s + (−0.207 − 0.978i)17-s + (0.866 − 0.5i)18-s + (0.669 + 0.743i)19-s + (0.951 − 0.309i)22-s + (0.406 + 0.913i)23-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.743 − 0.669i)3-s + (−0.669 + 0.743i)4-s + (−0.309 + 0.951i)6-s + (0.951 + 0.309i)8-s + (0.104 + 0.994i)9-s + (−0.104 + 0.994i)11-s + (0.994 − 0.104i)12-s + (0.587 + 0.809i)13-s + (−0.104 − 0.994i)16-s + (−0.207 − 0.978i)17-s + (0.866 − 0.5i)18-s + (0.669 + 0.743i)19-s + (0.951 − 0.309i)22-s + (0.406 + 0.913i)23-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.784 - 0.620i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.784 - 0.620i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6236890824 - 0.2168116137i\)
\(L(\frac12)\) \(\approx\) \(0.6236890824 - 0.2168116137i\)
\(L(1)\) \(\approx\) \(0.6292957526 - 0.2593990132i\)
\(L(1)\) \(\approx\) \(0.6292957526 - 0.2593990132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.406 - 0.913i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (-0.207 - 0.978i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.406 + 0.913i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.994 + 0.104i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (0.743 + 0.669i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.207 + 0.978i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.994 + 0.104i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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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

−27.428083366182271789426784288707, −26.59226119205659184155517100115, −25.87716091669002707759178308436, −24.51472440837119289053169094468, −23.83479011773450974687261625342, −22.78131173900782447420819506897, −22.07745313829796083632948479278, −20.91929418319557746529640309240, −19.59790447854706257190564653924, −18.40565292030331594161614685433, −17.59367905145371785712253859367, −16.677748243279785442899826598908, −15.842545770475150960505874735314, −15.11001686493614532424994592620, −13.88144943899246853131239168629, −12.67504619957662946203956080399, −11.02612275786525386550979603175, −10.44988379031005203527720609161, −9.13108733194786209975091691854, −8.24872882287284442947838489836, −6.71807619409649569383452282316, −5.81775210299252719022945815232, −4.88752535987385010258438163188, −3.49886693294422654481565682009, −0.83438079337319009422162164885, 1.23989856418604516245677320542, 2.40430605225596256632046130800, 4.125345513156283612421600643821, 5.33862295276711058537982370815, 6.93739934397269693983693376818, 7.84668584098711428839001198770, 9.279292591689503529503115574, 10.28798712329165285699741611139, 11.53222300324303305629449288324, 11.99317162462815291355131594774, 13.19434519995351833311668770459, 13.95934848684461389395626184005, 15.83196177485877228782654618080, 16.92654789732704120650181464470, 17.7963841682054045789594888377, 18.55100814942770886758213755859, 19.39312705421882838892267598712, 20.54625359300553801847973866847, 21.41685206570409877178163518679, 22.82823875650021612169260557998, 22.95674989036598875415694054941, 24.46987158154937749010344138546, 25.499510349868335891947563713266, 26.570950001612869504397145239105, 27.71427969255647128392824663552

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