↑

Properties

Label 1-77-77.62-r0-0-0
Degree $1$
Conductor $77$
Sign $0.970 + 0.242i$
Analytic cond. $0.357586$
Root an. cond. $0.357586$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.309 − 0.951i)6-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 10-s − 12-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + (−0.309 − 0.951i)20-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.309 − 0.951i)6-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 10-s − 12-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + (−0.309 − 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ 0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9253962889 + 0.1137557129i\)
\(L(\frac12)\) \(\approx\) \(0.9253962889 + 0.1137557129i\)
\(L(1)\) \(\approx\) \(1.002698698 + 0.01203219186i\)
\(L(1)\) \(\approx\) \(1.002698698 + 0.01203219186i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.309 - 0.951i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.538222400468231143810899208383, −30.42443348884388036841996546001, −28.92822377737561451560139256188, −27.73767545710150054447725860504, −26.83731241853718166219659007596, −25.4314766272151304582807867518, −25.0338030058481403668120072804, −23.83106560132621771141941927429, −23.1777963179850444887877354605, −21.25212008651949262988401973597, −19.956773815534323803331065623663, −19.1306803698067717449662233125, −17.88978956433101531855604445518, −16.8084491102901441329411572682, −15.53454283332391589832045452979, −14.625437373363294451467811832797, −13.258392005195958396966905611274, −12.506948868296037870440631061558, −10.23452340510790345656292173913, −8.68645455875435743786075968443, −8.25659010312455200290410986619, −6.88583560842524454803895712761, −5.38909528559575485125783434717, −3.758570455065874744064063042460, −1.29819979742838519096564895388, 2.282248912904032642846602661801, 3.39294880185787643734756846901, 4.602635637096723602231467780023, 7.12405617327406251662328325077, 8.50199632430993164355383741449, 9.60618238131905646650268917638, 10.70887303491955109306917120725, 11.67919097708322044763377292873, 13.42732391816309275199519676114, 14.31949230308043821784798757193, 15.57894174254666558758283981415, 17.03618584715542267861300445081, 18.674096590188287876530922457398, 19.15341074706601018623725108930, 20.393196028298884408641031088529, 21.36254921243849200169608963666, 22.239561785113222393484353028037, 23.394458859052862334893841779892, 25.36088189018762385266954973220, 26.26816374965476859381224275339, 27.01245545162468757232465720899, 27.90980369953970360685359540634, 29.28245762763062913806365275976, 30.39883770389528582097887302939, 31.119345800830812067339303635126

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