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Properties

Label 1-925-925.292-r1-0-0
Degree $1$
Conductor $925$
Sign $-0.862 + 0.506i$
Analytic cond. $99.4050$
Root an. cond. $99.4050$
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  + (0.275 + 0.961i)2-s + (0.999 − 0.0348i)3-s + (−0.848 + 0.529i)4-s + (0.309 + 0.951i)6-s + (0.984 + 0.173i)7-s + (−0.743 − 0.669i)8-s + (0.997 − 0.0697i)9-s + (0.913 + 0.406i)11-s + (−0.829 + 0.559i)12-s + (−0.970 + 0.241i)13-s + (0.104 + 0.994i)14-s + (0.438 − 0.898i)16-s + (−0.529 + 0.848i)17-s + (0.342 + 0.939i)18-s + (0.615 + 0.788i)19-s + ⋯
L(s)  = 1  + (0.275 + 0.961i)2-s + (0.999 − 0.0348i)3-s + (−0.848 + 0.529i)4-s + (0.309 + 0.951i)6-s + (0.984 + 0.173i)7-s + (−0.743 − 0.669i)8-s + (0.997 − 0.0697i)9-s + (0.913 + 0.406i)11-s + (−0.829 + 0.559i)12-s + (−0.970 + 0.241i)13-s + (0.104 + 0.994i)14-s + (0.438 − 0.898i)16-s + (−0.529 + 0.848i)17-s + (0.342 + 0.939i)18-s + (0.615 + 0.788i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.862 + 0.506i$
Analytic conductor: \(99.4050\)
Root analytic conductor: \(99.4050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 925,\ (1:\ ),\ -0.862 + 0.506i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9479491275 + 3.483493061i\)
\(L(\frac12)\) \(\approx\) \(0.9479491275 + 3.483493061i\)
\(L(1)\) \(\approx\) \(1.356878323 + 1.130268902i\)
\(L(1)\) \(\approx\) \(1.356878323 + 1.130268902i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.275 + 0.961i)T \)
3 \( 1 + (0.999 - 0.0348i)T \)
7 \( 1 + (0.984 + 0.173i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.970 + 0.241i)T \)
17 \( 1 + (-0.529 + 0.848i)T \)
19 \( 1 + (0.615 + 0.788i)T \)
23 \( 1 + (-0.994 + 0.104i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.241 - 0.970i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.743 + 0.669i)T \)
53 \( 1 + (0.139 - 0.990i)T \)
59 \( 1 + (-0.438 + 0.898i)T \)
61 \( 1 + (-0.997 - 0.0697i)T \)
67 \( 1 + (0.139 + 0.990i)T \)
71 \( 1 + (0.0348 + 0.999i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.882 + 0.469i)T \)
83 \( 1 + (0.529 - 0.848i)T \)
89 \( 1 + (0.719 + 0.694i)T \)
97 \( 1 + (-0.743 + 0.669i)T \)
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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

−21.35843152119784939312123723033, −20.31974643845558444477160756963, −19.96251201026158035579749575015, −19.35527080130767204782334540125, −18.22301189567189540650972776511, −17.82270077150367708795719418215, −16.65367955406232779852524371652, −15.3541069668330041773117793085, −14.73985448740187407091569885234, −13.856956199177861912919722622940, −13.66970020136160743059198570428, −12.37502434780319884102819602439, −11.69588583596035143342285636531, −10.896074198605978854510447394833, −9.80469641983163700186771951190, −9.29586579167451506691315713470, −8.35433692357168106966411307525, −7.59791924166988642608216929211, −6.37431502341676641959914576868, −4.850604773408292163272092317763, −4.513167020027844879248174459448, −3.34602721917268374327433678170, −2.53224170788913650929345145710, −1.64779620950176412086020394226, −0.60858789156247440972012596261, 1.34340978921349384110706509267, 2.34876843944721056550856586674, 3.7003404027358935817106532868, 4.343848590323024610767664943860, 5.20984249300566131518332496878, 6.444409373334731350016093381230, 7.2404457413113368039795044557, 8.053878751976686814147651417146, 8.63840578230361036102605228484, 9.509773759242880621100744145229, 10.33022087843539070877308803750, 12.01517389477015639287612353657, 12.32205544917309434426776339651, 13.61605014902317644456015378306, 14.254409529864579046648435398978, 14.683540960840178342139307598756, 15.42542179532855789661800988406, 16.26697349809658849029594983218, 17.38843897998446264162227291719, 17.77751703410387502974267995238, 18.811150849653034751924193467840, 19.63119763268740204723998039554, 20.43552717679611055383828190024, 21.451868224403491766610642657721, 21.87283535793877526927048743644

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