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Properties

Label 1-307-307.235-r0-0-0
Degree $1$
Conductor $307$
Sign $0.523 + 0.852i$
Analytic cond. $1.42570$
Root an. cond. $1.42570$
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 + (0.0922 + 0.995i)3-s + (0.932 + 0.361i)4-s + (0.932 − 0.361i)5-s + (0.0922 − 0.995i)6-s + (0.445 + 0.895i)7-s + (−0.850 − 0.526i)8-s + (−0.982 + 0.183i)9-s + (−0.982 + 0.183i)10-s + (0.0922 − 0.995i)11-s + (−0.273 + 0.961i)12-s + (0.739 + 0.673i)13-s + (−0.273 − 0.961i)14-s + (0.445 + 0.895i)15-s + (0.739 + 0.673i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.982 − 0.183i)2-s + (0.0922 + 0.995i)3-s + (0.932 + 0.361i)4-s + (0.932 − 0.361i)5-s + (0.0922 − 0.995i)6-s + (0.445 + 0.895i)7-s + (−0.850 − 0.526i)8-s + (−0.982 + 0.183i)9-s + (−0.982 + 0.183i)10-s + (0.0922 − 0.995i)11-s + (−0.273 + 0.961i)12-s + (0.739 + 0.673i)13-s + (−0.273 − 0.961i)14-s + (0.445 + 0.895i)15-s + (0.739 + 0.673i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(307\)
Sign: $0.523 + 0.852i$
Analytic conductor: \(1.42570\)
Root analytic conductor: \(1.42570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{307} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 307,\ (0:\ ),\ 0.523 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9272887410 + 0.5187572425i\)
\(L(\frac12)\) \(\approx\) \(0.9272887410 + 0.5187572425i\)
\(L(1)\) \(\approx\) \(0.8722266253 + 0.2632189619i\)
\(L(1)\) \(\approx\) \(0.8722266253 + 0.2632189619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad307 \( 1 \)
good2 \( 1 + (-0.982 - 0.183i)T \)
3 \( 1 + (0.0922 + 0.995i)T \)
5 \( 1 + (0.932 - 0.361i)T \)
7 \( 1 + (0.445 + 0.895i)T \)
11 \( 1 + (0.0922 - 0.995i)T \)
13 \( 1 + (0.739 + 0.673i)T \)
17 \( 1 + T \)
19 \( 1 + (0.739 + 0.673i)T \)
23 \( 1 + (-0.982 + 0.183i)T \)
29 \( 1 + (-0.982 - 0.183i)T \)
31 \( 1 + (0.0922 - 0.995i)T \)
37 \( 1 + (0.445 + 0.895i)T \)
41 \( 1 + (0.445 - 0.895i)T \)
43 \( 1 + (-0.850 - 0.526i)T \)
47 \( 1 + (0.739 + 0.673i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.602 + 0.798i)T \)
61 \( 1 + (-0.850 - 0.526i)T \)
67 \( 1 + (-0.273 + 0.961i)T \)
71 \( 1 + (-0.850 + 0.526i)T \)
73 \( 1 + (0.739 + 0.673i)T \)
79 \( 1 + (-0.850 + 0.526i)T \)
83 \( 1 + (0.445 - 0.895i)T \)
89 \( 1 + (-0.273 + 0.961i)T \)
97 \( 1 + (0.739 - 0.673i)T \)
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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

−25.27224250848737769030666453011, −24.54765566169409717336659146683, −23.52436337012955952509047540601, −22.79591076225488786383413831593, −21.23147150214432903357150356223, −20.25600996271380425264847913864, −19.82369402532331614570220170525, −18.29785943412646267637216139649, −18.10607414839770844862447586874, −17.31160045494018856950206472660, −16.443590755107490791267331253469, −14.94402044965830479592481475199, −14.19133705027831695675553270145, −13.25258537933504350776670040372, −12.09235489294577550498253816366, −10.96495269110958644044451204858, −10.14904913335124968044119294828, −9.15856762783865547261355281704, −7.87767060721848894886845126688, −7.29537736738959640578762603177, −6.33866374804163463871887687003, −5.353817685617654364880107188877, −3.18563037666353125357023826422, −1.917225704564389447092323630356, −1.100525188543149420612852928493, 1.44542493752709221244909420049, 2.64826860944925989385720668759, 3.845815453646384121428520766440, 5.64407329686784611986464649675, 5.95954141583621642499781108454, 7.94593109541849473762165590161, 8.77334038933612519915747720690, 9.44453722473925648426740262806, 10.262673569746568362805704523306, 11.35803929774269868049088883050, 12.03863201262337713640390408508, 13.65892311525758379037800638457, 14.586685864253386805765968537555, 15.74141844456874651555402779359, 16.50080467373240019266439217304, 17.097974462521392660293597704003, 18.345009479695714753562651619180, 18.85511155231491721689202604121, 20.313282084755306407041142573888, 20.941485049376724038203390339041, 21.539776610075226115316836929692, 22.25140923802747480320416572255, 24.01197298765692683247381408440, 24.82370871721042593110016312164, 25.71734785250271301805877603583

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