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Properties

Label 1-137-137.133-r0-0-0
Degree $1$
Conductor $137$
Sign $0.934 - 0.355i$
Analytic cond. $0.636225$
Root an. cond. $0.636225$
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  + (−0.982 + 0.183i)2-s + (−0.602 − 0.798i)3-s + (0.932 − 0.361i)4-s + (−0.982 − 0.183i)5-s + (0.739 + 0.673i)6-s + (0.445 + 0.895i)7-s + (−0.850 + 0.526i)8-s + (−0.273 + 0.961i)9-s + 10-s + (0.932 − 0.361i)11-s + (−0.850 − 0.526i)12-s + (0.445 + 0.895i)13-s + (−0.602 − 0.798i)14-s + (0.445 + 0.895i)15-s + (0.739 − 0.673i)16-s + (−0.850 − 0.526i)17-s + ⋯
L(s)  = 1  + (−0.982 + 0.183i)2-s + (−0.602 − 0.798i)3-s + (0.932 − 0.361i)4-s + (−0.982 − 0.183i)5-s + (0.739 + 0.673i)6-s + (0.445 + 0.895i)7-s + (−0.850 + 0.526i)8-s + (−0.273 + 0.961i)9-s + 10-s + (0.932 − 0.361i)11-s + (−0.850 − 0.526i)12-s + (0.445 + 0.895i)13-s + (−0.602 − 0.798i)14-s + (0.445 + 0.895i)15-s + (0.739 − 0.673i)16-s + (−0.850 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.934 - 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5180799753 - 0.09511034930i\)
\(L(\frac12)\) \(\approx\) \(0.5180799753 - 0.09511034930i\)
\(L(1)\) \(\approx\) \(0.5626573656 - 0.06600797267i\)
\(L(1)\) \(\approx\) \(0.5626573656 - 0.06600797267i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.982 + 0.183i)T \)
3 \( 1 + (-0.602 - 0.798i)T \)
5 \( 1 + (-0.982 - 0.183i)T \)
7 \( 1 + (0.445 + 0.895i)T \)
11 \( 1 + (0.932 - 0.361i)T \)
13 \( 1 + (0.445 + 0.895i)T \)
17 \( 1 + (-0.850 - 0.526i)T \)
19 \( 1 + (0.0922 - 0.995i)T \)
23 \( 1 + (0.739 - 0.673i)T \)
29 \( 1 + (0.739 - 0.673i)T \)
31 \( 1 + (0.0922 + 0.995i)T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + (0.0922 - 0.995i)T \)
47 \( 1 + (-0.273 + 0.961i)T \)
53 \( 1 + (0.0922 - 0.995i)T \)
59 \( 1 + (-0.273 + 0.961i)T \)
61 \( 1 + (-0.273 - 0.961i)T \)
67 \( 1 + (0.445 + 0.895i)T \)
71 \( 1 + (0.932 + 0.361i)T \)
73 \( 1 + (0.445 - 0.895i)T \)
79 \( 1 + (-0.602 + 0.798i)T \)
83 \( 1 + (-0.850 + 0.526i)T \)
89 \( 1 + (-0.982 - 0.183i)T \)
97 \( 1 + (0.932 - 0.361i)T \)
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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

−28.12721824684675948937479345229, −27.44322373731486378812473222371, −26.976939446696112899583447587701, −26.01657302019772617896406479740, −24.68002182694883561147591677664, −23.44344935487708590885982899000, −22.64676382177839490392121903425, −21.343081936090258013753613220418, −20.22273841285322560641506235138, −19.76667472524705114255805231661, −18.25183120111865579431325381963, −17.31659992368103061763501417926, −16.55041650426194742057291486934, −15.49543975348831856226425610383, −14.70167833697460426270821849717, −12.58612616859054108099017668914, −11.39241933357126896925846113139, −10.883844979931222124795259269873, −9.82764360152777515636981369525, −8.5238780775395621564357687739, −7.428642122329920929024203764282, −6.23739765864995738417213251765, −4.33684412437557111371226642798, −3.41532225425826693835256349231, −1.034556922895572079534200483264, 1.00016630660112949470569367669, 2.51807686749374243619083849884, 4.75793774299311384411349896414, 6.3103321426912488603719369674, 7.10184625876155439988218211428, 8.41805290067844041789749037983, 9.05083305189969189763581849153, 11.21712354550763707856108530271, 11.42080183125158949015855465917, 12.47777121262610301024881521551, 14.20598012915251512659530433482, 15.53197989005341482667364425099, 16.356107773756823855588786161391, 17.420626110069326857479051341898, 18.37532247502379319970552559595, 19.17993578480122666901949129498, 19.87363747608073650166094235067, 21.358766564422425555092417970676, 22.66626341255088303333940473828, 23.8974423614791016146896000947, 24.45394305059873561449297514773, 25.2559594488877667958343003598, 26.725466669588539540934873828756, 27.52288414528422979113617772138, 28.44382770752612774785916473415

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