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Properties

Label 1-137-137.43-r1-0-0
Degree $1$
Conductor $137$
Sign $0.972 + 0.233i$
Analytic cond. $14.7226$
Root an. cond. $14.7226$
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  + (0.673 + 0.739i)2-s + (−0.228 − 0.973i)3-s + (−0.0922 + 0.995i)4-s + (−0.998 + 0.0461i)5-s + (0.565 − 0.824i)6-s + (0.961 − 0.273i)7-s + (−0.798 + 0.602i)8-s + (−0.895 + 0.445i)9-s + (−0.707 − 0.707i)10-s + (0.995 + 0.0922i)11-s + (0.990 − 0.138i)12-s + (0.486 − 0.873i)13-s + (0.850 + 0.526i)14-s + (0.273 + 0.961i)15-s + (−0.982 − 0.183i)16-s + (0.798 + 0.602i)17-s + ⋯
L(s)  = 1  + (0.673 + 0.739i)2-s + (−0.228 − 0.973i)3-s + (−0.0922 + 0.995i)4-s + (−0.998 + 0.0461i)5-s + (0.565 − 0.824i)6-s + (0.961 − 0.273i)7-s + (−0.798 + 0.602i)8-s + (−0.895 + 0.445i)9-s + (−0.707 − 0.707i)10-s + (0.995 + 0.0922i)11-s + (0.990 − 0.138i)12-s + (0.486 − 0.873i)13-s + (0.850 + 0.526i)14-s + (0.273 + 0.961i)15-s + (−0.982 − 0.183i)16-s + (0.798 + 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.972 + 0.233i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ 0.972 + 0.233i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.239807850 + 0.2646423803i\)
\(L(\frac12)\) \(\approx\) \(2.239807850 + 0.2646423803i\)
\(L(1)\) \(\approx\) \(1.414100874 + 0.1933459728i\)
\(L(1)\) \(\approx\) \(1.414100874 + 0.1933459728i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (0.673 + 0.739i)T \)
3 \( 1 + (-0.228 - 0.973i)T \)
5 \( 1 + (-0.998 + 0.0461i)T \)
7 \( 1 + (0.961 - 0.273i)T \)
11 \( 1 + (0.995 + 0.0922i)T \)
13 \( 1 + (0.486 - 0.873i)T \)
17 \( 1 + (0.798 + 0.602i)T \)
19 \( 1 + (0.361 - 0.932i)T \)
23 \( 1 + (0.565 + 0.824i)T \)
29 \( 1 + (0.824 - 0.565i)T \)
31 \( 1 + (0.914 + 0.403i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.403 + 0.914i)T \)
47 \( 1 + (-0.948 - 0.317i)T \)
53 \( 1 + (-0.914 + 0.403i)T \)
59 \( 1 + (0.445 + 0.895i)T \)
61 \( 1 + (-0.895 - 0.445i)T \)
67 \( 1 + (0.873 + 0.486i)T \)
71 \( 1 + (-0.638 + 0.769i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (-0.973 - 0.228i)T \)
83 \( 1 + (0.138 - 0.990i)T \)
89 \( 1 + (-0.0461 - 0.998i)T \)
97 \( 1 + (-0.769 + 0.638i)T \)
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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

−28.049739322350253660638022910516, −27.44292132078450222807083716968, −26.80115395756132157780020711666, −24.992649588628015492636725617889, −23.87986702908887339170416650021, −22.99462603478470347184232595653, −22.22690202417013849176720033110, −21.03737720651107427053293303588, −20.59354432628470626404272399626, −19.37682355165469184278572854118, −18.38568235497489030907662293134, −16.76182289482762009054513995683, −15.77890210253135790659424331571, −14.64083260167666908738889479353, −14.17765069703795740080930426491, −12.10214255254671865019426993485, −11.66829597515153141085861350466, −10.78365720108892155921505571653, −9.456683615900892286659087181598, −8.358916620963546475529817400214, −6.40925777769314675749251185325, −4.957671908489287041756040522290, −4.21807155475242025578610653421, −3.14726047012108925313747569356, −1.12916767868239001775234532315, 1.017188210317394207715390471296, 3.11337997532709251796941065438, 4.45497154722978106428782010335, 5.72556796489295672987939698012, 7.01151973796341931417810585195, 7.80978323859650971378873423716, 8.61444039140432579328034549508, 11.121958733935109238262222670002, 11.8401081480857161756497069329, 12.80455077724581526731345099189, 13.98523489532984189255341445014, 14.821078310821058068731707907399, 15.92608857076174221638128912236, 17.27257575977810755767560475397, 17.76517328797273082525244494105, 19.22753451382028713490481264827, 20.187882274743364565182609310733, 21.49169413812453658450169253518, 22.93822658718594449959504888318, 23.22495497833920198925153402525, 24.32795147478271074454688827016, 24.87322449710457460509369296401, 26.06466712687394601047984337195, 27.30488504808299395683914758306, 28.05073172942964432613926048715

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