↑

Properties

Label 1-235-235.8-r1-0-0
Degree $1$
Conductor $235$
Sign $0.726 + 0.687i$
Analytic cond. $25.2542$
Root an. cond. $25.2542$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.730 − 0.682i)2-s + (−0.136 − 0.990i)3-s + (0.0682 − 0.997i)4-s + (−0.775 − 0.631i)6-s + (−0.398 + 0.917i)7-s + (−0.631 − 0.775i)8-s + (−0.962 + 0.269i)9-s + (0.203 + 0.979i)11-s + (−0.997 + 0.0682i)12-s + (0.519 + 0.854i)13-s + (0.334 + 0.942i)14-s + (−0.990 − 0.136i)16-s + (−0.979 − 0.203i)17-s + (−0.519 + 0.854i)18-s + (−0.460 + 0.887i)19-s + ⋯
L(s)  = 1  + (0.730 − 0.682i)2-s + (−0.136 − 0.990i)3-s + (0.0682 − 0.997i)4-s + (−0.775 − 0.631i)6-s + (−0.398 + 0.917i)7-s + (−0.631 − 0.775i)8-s + (−0.962 + 0.269i)9-s + (0.203 + 0.979i)11-s + (−0.997 + 0.0682i)12-s + (0.519 + 0.854i)13-s + (0.334 + 0.942i)14-s + (−0.990 − 0.136i)16-s + (−0.979 − 0.203i)17-s + (−0.519 + 0.854i)18-s + (−0.460 + 0.887i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(25.2542\)
Root analytic conductor: \(25.2542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (1:\ ),\ 0.726 + 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9064744401 + 0.3609330756i\)
\(L(\frac12)\) \(\approx\) \(0.9064744401 + 0.3609330756i\)
\(L(1)\) \(\approx\) \(1.004772986 - 0.4748246058i\)
\(L(1)\) \(\approx\) \(1.004772986 - 0.4748246058i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.730 - 0.682i)T \)
3 \( 1 + (-0.136 - 0.990i)T \)
7 \( 1 + (-0.398 + 0.917i)T \)
11 \( 1 + (0.203 + 0.979i)T \)
13 \( 1 + (0.519 + 0.854i)T \)
17 \( 1 + (-0.979 - 0.203i)T \)
19 \( 1 + (-0.460 + 0.887i)T \)
23 \( 1 + (0.730 + 0.682i)T \)
29 \( 1 + (-0.854 - 0.519i)T \)
31 \( 1 + (-0.990 - 0.136i)T \)
37 \( 1 + (0.942 + 0.334i)T \)
41 \( 1 + (-0.775 - 0.631i)T \)
43 \( 1 + (-0.997 - 0.0682i)T \)
53 \( 1 + (0.631 - 0.775i)T \)
59 \( 1 + (0.0682 + 0.997i)T \)
61 \( 1 + (-0.334 - 0.942i)T \)
67 \( 1 + (0.398 + 0.917i)T \)
71 \( 1 + (0.682 - 0.730i)T \)
73 \( 1 + (-0.269 + 0.962i)T \)
79 \( 1 + (0.576 + 0.816i)T \)
83 \( 1 + (-0.979 + 0.203i)T \)
89 \( 1 + (-0.460 - 0.887i)T \)
97 \( 1 + (0.136 + 0.990i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.03410557666039436068732994495, −25.037786302806373292388070234896, −23.8640931189034831364397420092, −23.13468997935845572189337118274, −22.21285386028668720227210627636, −21.63489306534340901134253788217, −20.48494225830191527108933398712, −19.85059594616819932297556599247, −18.07848683086555018534250706604, −16.93769271517126827389830606446, −16.487430889231578920978215109, −15.47848578924179055000028401540, −14.71747882446597998854515372219, −13.570030138858845489238738883805, −12.9050173384334199269694757234, −11.22803720267497618296719681890, −10.74930623726049675326476494402, −9.165677858720468174446006606829, −8.31458411190093668788392965093, −6.87351544026283051208369311522, −5.94162801318595588361269922855, −4.808276989630694191362444423974, −3.80966703643349909852158915650, −2.99049072728593227010038297504, −0.23567836818817833191277516360, 1.63714948746442334584248076397, 2.34547747303338163849203084976, 3.79605487510002490148452611479, 5.21428119255501090154767781900, 6.23602645090669934740562110560, 7.05202598673443817627807867037, 8.728726386051968950324673388076, 9.66597236044649466467784817098, 11.19577681647205798405969297541, 11.851245149668674882387660476455, 12.7828696364084909765088977681, 13.404932396253026437113740315582, 14.601176690692404186785584468756, 15.40436411738166041603215345466, 16.802081378563977130148670798, 18.185212507814145455708639102580, 18.73185489672450554828199592676, 19.62655751373139408415681780425, 20.5141020860683530380239971242, 21.640145090157534214429792902716, 22.56333387688719591742447935978, 23.2436521705408112545325138769, 24.154638121528246517216501784832, 25.06776743162277969905955942373, 25.7389169480224712486877836489

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