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Properties

Label 1-75-75.53-r0-0-0
Degree $1$
Conductor $75$
Sign $-0.0627 - 0.998i$
Analytic cond. $0.348298$
Root an. cond. $0.348298$
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  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s − i·7-s + (−0.951 − 0.309i)8-s + (0.809 + 0.587i)11-s + (−0.587 − 0.809i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.951 − 0.309i)22-s + (−0.587 + 0.809i)23-s − 26-s + (−0.951 + 0.309i)28-s + (0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s − i·7-s + (−0.951 − 0.309i)8-s + (0.809 + 0.587i)11-s + (−0.587 − 0.809i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.951 − 0.309i)22-s + (−0.587 + 0.809i)23-s − 26-s + (−0.951 + 0.309i)28-s + (0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.0627 - 0.998i$
Analytic conductor: \(0.348298\)
Root analytic conductor: \(0.348298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 75,\ (0:\ ),\ -0.0627 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8197428067 - 0.8729374261i\)
\(L(\frac12)\) \(\approx\) \(0.8197428067 - 0.8729374261i\)
\(L(1)\) \(\approx\) \(1.069220678 - 0.6848339633i\)
\(L(1)\) \(\approx\) \(1.069220678 - 0.6848339633i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
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

−31.97727783298704135429607163476, −30.80919962291163479456577059487, −29.849927875379280155763542933640, −28.39396836606520631025231010487, −27.168470256963202677939516079682, −26.10887860715022265493603194456, −24.93665999131596187234035370283, −24.31323850526920467966753862252, −23.02244478044875619180185605999, −21.87523493968284765637138787805, −21.28557584804028384867106129278, −19.46337050685427328010678090114, −18.24607542695499427982243990403, −16.9611813596413257524593468204, −16.00427409699980467651639812186, −14.78491995352173966135157533102, −13.934284636244440934024094714479, −12.42395712074309832391490482308, −11.602778671889718789309263302980, −9.397554135106862678166172291473, −8.37221803513104671845458144076, −6.83529837885524590789398278800, −5.7243118415526492651109797832, −4.352802393821713721570888435407, −2.6981146907434888292025472424, 1.44566632682402016700096103213, 3.35892202785829258183672622608, 4.52589114859574523808280762539, 6.06553392541314464229088960845, 7.69454124749723506138908074087, 9.65981197019267084471610942273, 10.45423156629288035399165757743, 11.86909039159191184846182086263, 12.86927195543700901091597192105, 14.14284011639769506078056792672, 14.981229825318942717924138260687, 16.73060876346770775889657270808, 17.93549414830576171058697093736, 19.422934613372098656108820822645, 20.13362911937981176572328934081, 21.19081965784646915882494518131, 22.4891925729003687858391994105, 23.208607751067389566741760230249, 24.353473481339426094874168467598, 25.67886734165918845629601735486, 27.296679516035336519295994930313, 27.82360671912937870887340642446, 29.47610693209203314914976500360, 29.86319151260367601245814556741, 30.98838738185896582132904602684

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