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Properties

Label 1-1275-1275.641-r0-0-0
Degree $1$
Conductor $1275$
Sign $0.348 + 0.937i$
Analytic cond. $5.92107$
Root an. cond. $5.92107$
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  + (0.891 + 0.453i)2-s + (0.587 + 0.809i)4-s + (0.923 − 0.382i)7-s + (0.156 + 0.987i)8-s + (−0.760 + 0.649i)11-s + (0.951 + 0.309i)13-s + (0.996 + 0.0784i)14-s + (−0.309 + 0.951i)16-s + (0.987 − 0.156i)19-s + (−0.972 + 0.233i)22-s + (0.760 − 0.649i)23-s + (0.707 + 0.707i)26-s + (0.852 + 0.522i)28-s + (−0.972 + 0.233i)29-s + (0.852 − 0.522i)31-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)2-s + (0.587 + 0.809i)4-s + (0.923 − 0.382i)7-s + (0.156 + 0.987i)8-s + (−0.760 + 0.649i)11-s + (0.951 + 0.309i)13-s + (0.996 + 0.0784i)14-s + (−0.309 + 0.951i)16-s + (0.987 − 0.156i)19-s + (−0.972 + 0.233i)22-s + (0.760 − 0.649i)23-s + (0.707 + 0.707i)26-s + (0.852 + 0.522i)28-s + (−0.972 + 0.233i)29-s + (0.852 − 0.522i)31-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $0.348 + 0.937i$
Analytic conductor: \(5.92107\)
Root analytic conductor: \(5.92107\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1275,\ (0:\ ),\ 0.348 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.569069858 + 1.784867604i\)
\(L(\frac12)\) \(\approx\) \(2.569069858 + 1.784867604i\)
\(L(1)\) \(\approx\) \(1.857828029 + 0.7312352835i\)
\(L(1)\) \(\approx\) \(1.857828029 + 0.7312352835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.891 + 0.453i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (-0.760 + 0.649i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.987 - 0.156i)T \)
23 \( 1 + (0.760 - 0.649i)T \)
29 \( 1 + (-0.972 + 0.233i)T \)
31 \( 1 + (0.852 - 0.522i)T \)
37 \( 1 + (-0.760 - 0.649i)T \)
41 \( 1 + (0.0784 + 0.996i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.156 + 0.987i)T \)
59 \( 1 + (-0.891 + 0.453i)T \)
61 \( 1 + (-0.649 - 0.760i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (-0.233 - 0.972i)T \)
73 \( 1 + (-0.0784 + 0.996i)T \)
79 \( 1 + (0.852 + 0.522i)T \)
83 \( 1 + (0.987 - 0.156i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (0.972 - 0.233i)T \)
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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

−20.88684392744279901714490740736, −20.52774618281973979724830193435, −19.4358488782793952103881984119, −18.616660340839316692198711382920, −18.13361201826417668394662616724, −17.01986495227614983684754434067, −15.90100631056589676464688748448, −15.48267338073995310017478721578, −14.66107224425928888098778560913, −13.67393621546193442129264361422, −13.414094095385071065550307024331, −12.29738788397255246684618807658, −11.53978687505430468981006372602, −10.975266858800428507504394572144, −10.23896635080295762679165867182, −9.12333657248043452501739544762, −8.19427665953564470938407795385, −7.34423958624635521507178030247, −6.19617725235209061913411332361, −5.382607484887378531273574319967, −4.937657323139635781940259254649, −3.65054623592608350491507812904, −3.03138998509596810134797793954, −1.9055917447312575924366559395, −1.01912195369507579109905635095, 1.35062123887105837380986814410, 2.38501377773398472925816135, 3.40864487547153705373024632884, 4.38286625534193201439537657073, 5.00780846300909756606795863558, 5.83352618700089988733482507404, 6.88389492926458142636620299839, 7.58542818446203616385475021988, 8.256042754372286516863848802219, 9.25414606213076477928953017067, 10.58456004740335537726297569909, 11.13697287626235996127080206942, 11.96043458505927825424326505200, 12.82673216184420030094598623249, 13.61598461850160201015058091057, 14.14542387084053213240007681331, 15.095694802985683427709846257308, 15.59156500777307913544181297390, 16.51167273279568095638953986948, 17.21801649101785385576391008576, 18.02953208707908028406410344725, 18.69925693171688528608363202371, 20.109952943975363459539711755448, 20.63396382357448928098159456093, 21.087914819504188341502902850532

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