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Properties

Label 1-1625-1625.1096-r0-0-0
Degree $1$
Conductor $1625$
Sign $-0.699 + 0.714i$
Analytic cond. $7.54646$
Root an. cond. $7.54646$
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  + (−0.228 + 0.973i)2-s + (−0.999 − 0.0418i)3-s + (−0.895 − 0.444i)4-s + (0.268 − 0.963i)6-s + (0.978 + 0.207i)7-s + (0.637 − 0.770i)8-s + (0.996 + 0.0836i)9-s + (−0.228 + 0.973i)11-s + (0.876 + 0.481i)12-s + (−0.425 + 0.904i)14-s + (0.604 + 0.796i)16-s + (0.832 + 0.553i)17-s + (−0.309 + 0.951i)18-s + (0.999 − 0.0418i)19-s + (−0.968 − 0.248i)21-s + (−0.895 − 0.444i)22-s + ⋯
L(s)  = 1  + (−0.228 + 0.973i)2-s + (−0.999 − 0.0418i)3-s + (−0.895 − 0.444i)4-s + (0.268 − 0.963i)6-s + (0.978 + 0.207i)7-s + (0.637 − 0.770i)8-s + (0.996 + 0.0836i)9-s + (−0.228 + 0.973i)11-s + (0.876 + 0.481i)12-s + (−0.425 + 0.904i)14-s + (0.604 + 0.796i)16-s + (0.832 + 0.553i)17-s + (−0.309 + 0.951i)18-s + (0.999 − 0.0418i)19-s + (−0.968 − 0.248i)21-s + (−0.895 − 0.444i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1625\)    =    \(5^{3} \cdot 13\)
Sign: $-0.699 + 0.714i$
Analytic conductor: \(7.54646\)
Root analytic conductor: \(7.54646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1625} (1096, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1625,\ (0:\ ),\ -0.699 + 0.714i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3859673268 + 0.9185107556i\)
\(L(\frac12)\) \(\approx\) \(0.3859673268 + 0.9185107556i\)
\(L(1)\) \(\approx\) \(0.6299272507 + 0.4414792692i\)
\(L(1)\) \(\approx\) \(0.6299272507 + 0.4414792692i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.228 + 0.973i)T \)
3 \( 1 + (-0.999 - 0.0418i)T \)
7 \( 1 + (0.978 + 0.207i)T \)
11 \( 1 + (-0.228 + 0.973i)T \)
17 \( 1 + (0.832 + 0.553i)T \)
19 \( 1 + (0.999 - 0.0418i)T \)
23 \( 1 + (-0.756 + 0.653i)T \)
29 \( 1 + (0.783 + 0.621i)T \)
31 \( 1 + (-0.0627 + 0.998i)T \)
37 \( 1 + (-0.604 - 0.796i)T \)
41 \( 1 + (-0.944 + 0.328i)T \)
43 \( 1 + (-0.104 - 0.994i)T \)
47 \( 1 + (0.637 + 0.770i)T \)
53 \( 1 + (0.968 + 0.248i)T \)
59 \( 1 + (0.855 - 0.518i)T \)
61 \( 1 + (0.944 + 0.328i)T \)
67 \( 1 + (-0.146 - 0.989i)T \)
71 \( 1 + (-0.985 + 0.166i)T \)
73 \( 1 + (-0.876 + 0.481i)T \)
79 \( 1 + (0.535 - 0.844i)T \)
83 \( 1 + (-0.535 - 0.844i)T \)
89 \( 1 + (0.855 + 0.518i)T \)
97 \( 1 + (-0.146 + 0.989i)T \)
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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

−20.43988256258182205042914416125, −19.26421726541874246154677102977, −18.515132500554728853349329536114, −18.12994956681378290208008267128, −17.325684406914077573422841285218, −16.63736151952762810617714366747, −16.01062354203089887992849386335, −14.79186046151008212129023505660, −13.83055728910284300335683264618, −13.39580392283909458940209675826, −12.197942728736851050456597156367, −11.68831663725719402564303663016, −11.27244548965051049281899911655, −10.2309639149797560968204561836, −9.96535451790709671111684081553, −8.661172754851420162602835404573, −7.97949976517831340280849475377, −7.14848127221663652224983771333, −5.811576633802039363332249882996, −5.213696333624310327368442394994, −4.40299818058899156237478345858, −3.54320049880613986453665894208, −2.432025834639183521888860899722, −1.29498451452747967605852230060, −0.59425125962171164316993411104, 1.092853784346547022780875888274, 1.83668892421930765234253725342, 3.66895185140100500720039884574, 4.63912378933517516909731553508, 5.28245968812592381869343905524, 5.77207898854542542731445221486, 6.93315564554056819782478193590, 7.43678549583800389126613538898, 8.21082468627177669190757734880, 9.204242570432906499371786357940, 10.19323827019741379389649719080, 10.55405713742371885288588119226, 11.86371439007499708780369462031, 12.25208748153772082595408498283, 13.287544128682274613800681041862, 14.19104700829755013326713625496, 14.8382246650458821465496323814, 15.738303454570692453008735261509, 16.14055080016544126156525617799, 17.21683942741552696919079653434, 17.62428408467991131547345032193, 18.14702387141681788121474653166, 18.82864898613003007117774515874, 19.85075993609524151582544227723, 20.85673518102281647544156916924

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