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Properties

Label 1-76-76.71-r0-0-0
Degree $1$
Conductor $76$
Sign $0.992 + 0.120i$
Analytic cond. $0.352942$
Root an. cond. $0.352942$
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  + (−0.939 − 0.342i)3-s + (0.173 + 0.984i)5-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (0.939 − 0.342i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)17-s + (−0.766 + 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)27-s + (−0.766 − 0.642i)29-s + (−0.5 + 0.866i)31-s + (−0.173 − 0.984i)33-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)3-s + (0.173 + 0.984i)5-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (0.939 − 0.342i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)17-s + (−0.766 + 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)27-s + (−0.766 − 0.642i)29-s + (−0.5 + 0.866i)31-s + (−0.173 − 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(0.352942\)
Root analytic conductor: \(0.352942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 76,\ (0:\ ),\ 0.992 + 0.120i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8135824987 + 0.04903977352i\)
\(L(\frac12)\) \(\approx\) \(0.8135824987 + 0.04903977352i\)
\(L(1)\) \(\approx\) \(0.8890535618 + 0.01835378084i\)
\(L(1)\) \(\approx\) \(0.8890535618 + 0.01835378084i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−31.49677583617361009120981768050, −30.074846011114636174964858939186, −28.927597566204550918191750199662, −28.012025049164564742272277983332, −27.53566993659275931931653835709, −25.93230084685762257602500197222, −24.4711865432204264672264276805, −23.94658566681937993078907446017, −22.51610340372847266074200960764, −21.40359792139792752758181316974, −20.79903745186219519179695047777, −19.00817176689478563983534589986, −17.917253339658016599352929741488, −16.70317037301703992759909234989, −16.06512154665685059092719617992, −14.60715654847674236264645419017, −12.95508117444841774265126700550, −11.93221427076666262092298207059, −10.946536249349562587317372924884, −9.342121317119082885771471483366, −8.338402880522388872657769401893, −6.17975227323914479848500385497, −5.36491213542093143253674175452, −3.96460373978699684551707954807, −1.39845297491799363830659869704, 1.58702375544705554397572365317, 3.78841354528954967058726514752, 5.430224701022469086446899090009, 6.80925897549110393407668749628, 7.62785468129583486173104795162, 9.88861948418508800898628498257, 10.8784468559433808113359796931, 11.80809742068884956155103291722, 13.33452569963245920101842441130, 14.41342742065765652512589367934, 15.83534712446400949601791846478, 17.24886603199047623073591687519, 17.915022662519050517483964441084, 18.976383269193511961462989099337, 20.45657833054174825351098652330, 21.750625251265996760353467417601, 23.01099611950523159293441405803, 23.2972217558275348145847923915, 24.85565164898485520735629335208, 25.968040538201862069060932334053, 27.28338056096053307003648750105, 28.03398395302858755309407811843, 29.53707602879429471948455519356, 30.08166788066724711132897728453, 30.93203897624112577887447555378

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