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Properties

Label 1-5025-5025.1856-r1-0-0
Degree $1$
Conductor $5025$
Sign $-0.995 - 0.0903i$
Analytic cond. $540.011$
Root an. cond. $540.011$
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  + (−0.548 − 0.836i)2-s + (−0.398 + 0.917i)4-s + (−0.888 + 0.458i)7-s + (0.985 − 0.170i)8-s + (−0.820 − 0.572i)11-s + (0.999 − 0.0380i)13-s + (0.870 + 0.491i)14-s + (−0.683 − 0.730i)16-s + (0.398 + 0.917i)17-s + (0.161 − 0.986i)19-s + (−0.0285 + 0.999i)22-s + (0.761 + 0.647i)23-s + (−0.580 − 0.814i)26-s + (−0.0665 − 0.997i)28-s + (−0.669 − 0.743i)29-s + ⋯
L(s)  = 1  + (−0.548 − 0.836i)2-s + (−0.398 + 0.917i)4-s + (−0.888 + 0.458i)7-s + (0.985 − 0.170i)8-s + (−0.820 − 0.572i)11-s + (0.999 − 0.0380i)13-s + (0.870 + 0.491i)14-s + (−0.683 − 0.730i)16-s + (0.398 + 0.917i)17-s + (0.161 − 0.986i)19-s + (−0.0285 + 0.999i)22-s + (0.761 + 0.647i)23-s + (−0.580 − 0.814i)26-s + (−0.0665 − 0.997i)28-s + (−0.669 − 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5025\)    =    \(3 \cdot 5^{2} \cdot 67\)
Sign: $-0.995 - 0.0903i$
Analytic conductor: \(540.011\)
Root analytic conductor: \(540.011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5025} (1856, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5025,\ (1:\ ),\ -0.995 - 0.0903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02319624115 - 0.5123532785i\)
\(L(\frac12)\) \(\approx\) \(0.02319624115 - 0.5123532785i\)
\(L(1)\) \(\approx\) \(0.6396090935 - 0.2209844506i\)
\(L(1)\) \(\approx\) \(0.6396090935 - 0.2209844506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.548 - 0.836i)T \)
7 \( 1 + (-0.888 + 0.458i)T \)
11 \( 1 + (-0.820 - 0.572i)T \)
13 \( 1 + (0.999 - 0.0380i)T \)
17 \( 1 + (0.398 + 0.917i)T \)
19 \( 1 + (0.161 - 0.986i)T \)
23 \( 1 + (0.761 + 0.647i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (0.345 - 0.938i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (-0.00951 + 0.999i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (0.432 + 0.901i)T \)
53 \( 1 + (-0.993 + 0.113i)T \)
59 \( 1 + (-0.696 + 0.717i)T \)
61 \( 1 + (-0.290 - 0.956i)T \)
71 \( 1 + (0.398 - 0.917i)T \)
73 \( 1 + (-0.290 - 0.956i)T \)
79 \( 1 + (0.640 - 0.768i)T \)
83 \( 1 + (-0.123 - 0.992i)T \)
89 \( 1 + (-0.610 + 0.791i)T \)
97 \( 1 + (-0.978 + 0.207i)T \)
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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

−18.206661624796831872113896727954, −17.555744702782834579963585420779, −16.63549067917620823261381914190, −16.26899643124522370727871369985, −15.7529746416093307063398823967, −15.06909749644728476384869816477, −14.11396570401999832570656379354, −13.8356403880171123132977631451, −12.82440385194775159551239821919, −12.49188392296888297095416569862, −11.14791085785328578381134326205, −10.61776840645857171876727014101, −9.98248808530426530087222192396, −9.36681347970197014068342495000, −8.663586834003489825949957164250, −7.90636237052642256785277584654, −7.13759679265656234094763903663, −6.791277800957914094640408216964, −5.79700500933369578059789489123, −5.32573168094114577874942361274, −4.39486386465080493492106199087, −3.572395643017155229035175567235, −2.68822204661375974631055502585, −1.5462223624024880621946001749, −0.75963260957056414986617261584, 0.12974356812961019340281442446, 0.93276062955685981516554951994, 1.85421296786365372669477143146, 2.85239859415282931406162151232, 3.21129127078749346571926965838, 4.01386739472990089267296932737, 4.97123890370728938871803150446, 5.89838727197863949521546836707, 6.47353679240243446668226532203, 7.60719097722687339045545364589, 8.075403021237419362639832210463, 8.95014471411842902643303142273, 9.362932957958639970396942252922, 10.15524733481254418647296498211, 10.87009285492974493951147989296, 11.32813654892385127924467341521, 12.137666691408048012912508588259, 12.93068066574869664704210768920, 13.3298698260787802106608063759, 13.79826062553090462134977613261, 15.23514919223194299167038155559, 15.53694988971744812709638700406, 16.38493735948956965720357787492, 16.964606815192361123072565138908, 17.656153946556597826977717022228

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