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Properties

Label 1-1976-1976.1051-r0-0-0
Degree $1$
Conductor $1976$
Sign $0.980 + 0.196i$
Analytic cond. $9.17650$
Root an. cond. $9.17650$
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 + (0.342 + 0.939i)5-s + (−0.866 − 0.5i)7-s + (0.766 − 0.642i)9-s + (−0.866 + 0.5i)11-s + (−0.642 − 0.766i)15-s + (0.939 − 0.342i)17-s + (0.984 + 0.173i)21-s + (0.766 − 0.642i)23-s + (−0.766 + 0.642i)25-s + (−0.5 + 0.866i)27-s + (0.939 + 0.342i)29-s + (−0.866 + 0.5i)31-s + (0.642 − 0.766i)33-s + (0.173 − 0.984i)35-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (0.342 + 0.939i)5-s + (−0.866 − 0.5i)7-s + (0.766 − 0.642i)9-s + (−0.866 + 0.5i)11-s + (−0.642 − 0.766i)15-s + (0.939 − 0.342i)17-s + (0.984 + 0.173i)21-s + (0.766 − 0.642i)23-s + (−0.766 + 0.642i)25-s + (−0.5 + 0.866i)27-s + (0.939 + 0.342i)29-s + (−0.866 + 0.5i)31-s + (0.642 − 0.766i)33-s + (0.173 − 0.984i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1976\)    =    \(2^{3} \cdot 13 \cdot 19\)
Sign: $0.980 + 0.196i$
Analytic conductor: \(9.17650\)
Root analytic conductor: \(9.17650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1976} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1976,\ (0:\ ),\ 0.980 + 0.196i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8753627530 + 0.08694238838i\)
\(L(\frac12)\) \(\approx\) \(0.8753627530 + 0.08694238838i\)
\(L(1)\) \(\approx\) \(0.7291165377 + 0.1249637056i\)
\(L(1)\) \(\approx\) \(0.7291165377 + 0.1249637056i\)

Euler product

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

−19.79700420611050556770255017763, −19.115596024056604475284597799714, −18.48876720341382818198037393885, −17.749586912517118381803119553, −16.91479089325115754307009833062, −16.41028358465195171263349584973, −15.87320624939385989147671628619, −15.04078913861752516765797649605, −13.721596176213249865781639930, −13.09887065694898053498073551136, −12.695680676763653278729225903326, −11.91036380629371639248972262951, −11.2182762313222600475121657610, −10.061481325301891373908734516793, −9.79940766828622018585360551485, −8.608200967975437634803078900394, −7.9882016770780160372702024391, −6.91956947172933206866586234196, −6.12370338644043040707245070420, −5.40064908239614644547176200941, −5.030933729577149318241189852259, −3.759127618888637914759107331435, −2.73205905051488340758485741697, −1.627095519808846789806813121516, −0.70179298929071060670720817053, 0.53418983460024656262387549205, 1.90642186014465481211646614526, 3.09347224391872243631089091728, 3.62480275059301035326580121348, 4.86283284654222207565226773143, 5.46745688778984039331013324710, 6.43301802053660197060239110017, 6.97638522327601659392536049011, 7.59978191802105096563439879849, 9.03102866972606044196984354476, 9.86480145210066428831709479486, 10.533764348216508883138615091574, 10.68157801965909076019171564667, 11.91812816966140804489338504739, 12.53656952289109786397936843763, 13.317056357937303341720360658367, 14.155839803885818517148674303353, 15.01206552723924449613685260214, 15.71349292255559079553305940092, 16.39687908962481840308921169864, 17.06141530313130122202392119308, 17.85737005773446511093336169084, 18.46032753445810257774449264541, 19.037322905652197287755468345484, 20.03168660974831730724714262786

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