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Properties

Label 1-373-373.77-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.807 + 0.589i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (0.363 + 0.931i)2-s + (−0.890 + 0.455i)3-s + (−0.736 + 0.676i)4-s + (−0.0337 + 0.999i)5-s + (−0.747 − 0.664i)6-s + (0.250 − 0.968i)7-s + (−0.897 − 0.440i)8-s + (0.585 − 0.810i)9-s + (−0.943 + 0.331i)10-s + (0.0675 − 0.997i)11-s + (0.347 − 0.937i)12-s + (0.440 + 0.897i)13-s + (0.993 − 0.117i)14-s + (−0.425 − 0.905i)15-s + (0.0843 − 0.996i)16-s + (0.954 + 0.299i)17-s + ⋯
L(s)  = 1  + (0.363 + 0.931i)2-s + (−0.890 + 0.455i)3-s + (−0.736 + 0.676i)4-s + (−0.0337 + 0.999i)5-s + (−0.747 − 0.664i)6-s + (0.250 − 0.968i)7-s + (−0.897 − 0.440i)8-s + (0.585 − 0.810i)9-s + (−0.943 + 0.331i)10-s + (0.0675 − 0.997i)11-s + (0.347 − 0.937i)12-s + (0.440 + 0.897i)13-s + (0.993 − 0.117i)14-s + (−0.425 − 0.905i)15-s + (0.0843 − 0.996i)16-s + (0.954 + 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.807 + 0.589i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.807 + 0.589i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5085261559 + 1.559143948i\)
\(L(\frac12)\) \(\approx\) \(0.5085261559 + 1.559143948i\)
\(L(1)\) \(\approx\) \(0.7346260416 + 0.6963182126i\)
\(L(1)\) \(\approx\) \(0.7346260416 + 0.6963182126i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.363 + 0.931i)T \)
3 \( 1 + (-0.890 + 0.455i)T \)
5 \( 1 + (-0.0337 + 0.999i)T \)
7 \( 1 + (0.250 - 0.968i)T \)
11 \( 1 + (0.0675 - 0.997i)T \)
13 \( 1 + (0.440 + 0.897i)T \)
17 \( 1 + (0.954 + 0.299i)T \)
19 \( 1 + (0.571 + 0.820i)T \)
23 \( 1 + (0.998 - 0.0506i)T \)
29 \( 1 + (0.283 + 0.959i)T \)
31 \( 1 + (-0.347 - 0.937i)T \)
37 \( 1 + (-0.990 + 0.134i)T \)
41 \( 1 + (0.979 + 0.201i)T \)
43 \( 1 + (0.676 + 0.736i)T \)
47 \( 1 + (0.598 - 0.801i)T \)
53 \( 1 + (0.810 - 0.585i)T \)
59 \( 1 + (0.972 + 0.234i)T \)
61 \( 1 + (-0.455 - 0.890i)T \)
67 \( 1 + (-0.848 - 0.528i)T \)
71 \( 1 + (-0.217 + 0.975i)T \)
73 \( 1 + (0.409 - 0.912i)T \)
79 \( 1 + (-0.331 - 0.943i)T \)
83 \( 1 + (0.585 + 0.810i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.299 + 0.954i)T \)
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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

−23.86745525160467217945073426834, −22.97782780203102983581660785462, −22.44326274491310161963793582652, −21.285598625036123442688090530108, −20.75892785281404325213223355244, −19.6736423688054198872253222258, −18.8304332643685884270294520645, −17.80650141228593337788864844679, −17.43091688512976534168912612036, −15.95998456423974849616181804423, −15.19555714655438803428360906065, −13.80336833170690577678490181579, −12.81243564179260959923689966475, −12.32362135690618276870199611048, −11.67793997174917631326019188378, −10.617427772508095327101293893573, −9.55444770789152096476110312636, −8.63638297126098456674241301958, −7.374037495707511352926711886850, −5.715868317417105839316229166125, −5.294472745426053156267110073427, −4.38878203685431935187539375750, −2.74895032116567700510880289824, −1.52592080001037808152190634806, −0.64388764308628968636705705414, 0.94725263838982065785979631686, 3.4212734702849712799816694872, 3.93991862277678697453915482368, 5.25663938531078636315616580689, 6.16126552316182055193091875199, 6.936543846755310642019360558653, 7.813221556292027724491831978670, 9.20116892773743992633670093487, 10.33183362483786431866125977140, 11.13649364936589645726807483447, 12.037989694125984868249854969316, 13.375894667030793892618663520590, 14.25162808697087243378616888729, 14.84922406474130442137840387645, 16.16489368163196430626110061937, 16.54720868483690900863819263614, 17.42936649161092445249351373912, 18.38617285210213469200628136106, 19.06300148826551553127648926531, 20.91562746223471760159317612241, 21.44131850009647193396487775397, 22.40313041726736515121791150989, 23.09549311337406588977306808470, 23.6532495268017435593712594304, 24.468430173928552909493958364607

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