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Properties

Label 1-149-149.30-r0-0-0
Degree $1$
Conductor $149$
Sign $0.989 + 0.141i$
Analytic cond. $0.691953$
Root an. cond. $0.691953$
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  + (−0.292 + 0.956i)2-s + (0.660 − 0.750i)3-s + (−0.828 − 0.559i)4-s + (0.873 − 0.487i)5-s + (0.524 + 0.851i)6-s + (0.210 + 0.977i)7-s + (0.778 − 0.628i)8-s + (−0.127 − 0.991i)9-s + (0.210 + 0.977i)10-s + (−0.828 + 0.559i)11-s + (−0.967 + 0.251i)12-s + (0.0424 − 0.999i)13-s + (−0.996 − 0.0848i)14-s + (0.210 − 0.977i)15-s + (0.372 + 0.927i)16-s + (0.660 − 0.750i)17-s + ⋯
L(s)  = 1  + (−0.292 + 0.956i)2-s + (0.660 − 0.750i)3-s + (−0.828 − 0.559i)4-s + (0.873 − 0.487i)5-s + (0.524 + 0.851i)6-s + (0.210 + 0.977i)7-s + (0.778 − 0.628i)8-s + (−0.127 − 0.991i)9-s + (0.210 + 0.977i)10-s + (−0.828 + 0.559i)11-s + (−0.967 + 0.251i)12-s + (0.0424 − 0.999i)13-s + (−0.996 − 0.0848i)14-s + (0.210 − 0.977i)15-s + (0.372 + 0.927i)16-s + (0.660 − 0.750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.989 + 0.141i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.989 + 0.141i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.233975001 + 0.08795685728i\)
\(L(\frac12)\) \(\approx\) \(1.233975001 + 0.08795685728i\)
\(L(1)\) \(\approx\) \(1.155166788 + 0.1381600722i\)
\(L(1)\) \(\approx\) \(1.155166788 + 0.1381600722i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.292 + 0.956i)T \)
3 \( 1 + (0.660 - 0.750i)T \)
5 \( 1 + (0.873 - 0.487i)T \)
7 \( 1 + (0.210 + 0.977i)T \)
11 \( 1 + (-0.828 + 0.559i)T \)
13 \( 1 + (0.0424 - 0.999i)T \)
17 \( 1 + (0.660 - 0.750i)T \)
19 \( 1 + (0.985 - 0.169i)T \)
23 \( 1 + (0.0424 + 0.999i)T \)
29 \( 1 + (-0.450 - 0.892i)T \)
31 \( 1 + (0.0424 + 0.999i)T \)
37 \( 1 + (-0.828 + 0.559i)T \)
41 \( 1 + (0.372 + 0.927i)T \)
43 \( 1 + (-0.594 - 0.803i)T \)
47 \( 1 + (0.985 + 0.169i)T \)
53 \( 1 + (-0.594 + 0.803i)T \)
59 \( 1 + (0.942 - 0.333i)T \)
61 \( 1 + (-0.292 + 0.956i)T \)
67 \( 1 + (-0.996 + 0.0848i)T \)
71 \( 1 + (0.873 - 0.487i)T \)
73 \( 1 + (-0.911 - 0.411i)T \)
79 \( 1 + (-0.911 + 0.411i)T \)
83 \( 1 + (-0.911 - 0.411i)T \)
89 \( 1 + (-0.721 + 0.691i)T \)
97 \( 1 + (-0.594 + 0.803i)T \)
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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

−28.172600204488674644572009611849, −26.89286521627558063117283816150, −26.339704025198530678004216184223, −25.80956238977115644080336274569, −24.160589980734774129745796930371, −22.76554757717340808668143624981, −21.81556104754452711181744325916, −21.008765153935215871885690452308, −20.476059067449903287054747086071, −19.19462440730839691612225664005, −18.40851426085112681815958613796, −17.08870113442703596151285219561, −16.31587547927717217665353706336, −14.43756605477546107151550813231, −13.9436947836520996576975648421, −12.98126538353836420497124699870, −11.18037239091543364495946993810, −10.44424475257960641722424350187, −9.721014094096976395237368648544, −8.580725939530080321312923758128, −7.38994192761134892662907915094, −5.37346125425876055714028167581, −4.04121069608450085541482798257, −3.00721571137720507383233876094, −1.74021903901021512082273198760, 1.37379871944806137987528772551, 2.85352449416771080106939976840, 5.15737513384222961976322190396, 5.795549720774507310352729663566, 7.289311822283284137463026948186, 8.16776738152482030377114117117, 9.20175448712806581000068699911, 9.99904568365883361575340146858, 12.15380045396388310980141072891, 13.17563065190598033103799268888, 13.92294675270191102072173263938, 15.08861696981571077123481655947, 15.8724822979509870279704119397, 17.46325090115268383278595041205, 18.033834079455858765622628509901, 18.77123961051263338976300903471, 20.14544856069242454700384267857, 21.1165840009665468641392682515, 22.466771267694785320921932265425, 23.60523587046337375007968891824, 24.6026570408800310535354537578, 25.24077989972935386069874600609, 25.67802067458909346750898823881, 26.93870484478400630140060925500, 28.18121034723455136849062888034

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