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Properties

Label 1-935-935.244-r0-0-0
Degree $1$
Conductor $935$
Sign $-0.357 - 0.934i$
Analytic cond. $4.34212$
Root an. cond. $4.34212$
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  + (−0.156 − 0.987i)2-s + (0.0784 + 0.996i)3-s + (−0.951 + 0.309i)4-s + (0.972 − 0.233i)6-s + (−0.996 − 0.0784i)7-s + (0.453 + 0.891i)8-s + (−0.987 + 0.156i)9-s + (−0.382 − 0.923i)12-s + (−0.587 + 0.809i)13-s + (0.0784 + 0.996i)14-s + (0.809 − 0.587i)16-s + (0.309 + 0.951i)18-s + (−0.891 + 0.453i)19-s − i·21-s + (−0.923 − 0.382i)23-s + (−0.852 + 0.522i)24-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (0.0784 + 0.996i)3-s + (−0.951 + 0.309i)4-s + (0.972 − 0.233i)6-s + (−0.996 − 0.0784i)7-s + (0.453 + 0.891i)8-s + (−0.987 + 0.156i)9-s + (−0.382 − 0.923i)12-s + (−0.587 + 0.809i)13-s + (0.0784 + 0.996i)14-s + (0.809 − 0.587i)16-s + (0.309 + 0.951i)18-s + (−0.891 + 0.453i)19-s − i·21-s + (−0.923 − 0.382i)23-s + (−0.852 + 0.522i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $-0.357 - 0.934i$
Analytic conductor: \(4.34212\)
Root analytic conductor: \(4.34212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 935,\ (0:\ ),\ -0.357 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2526524921 - 0.3671330497i\)
\(L(\frac12)\) \(\approx\) \(0.2526524921 - 0.3671330497i\)
\(L(1)\) \(\approx\) \(0.6429863756 - 0.1086278092i\)
\(L(1)\) \(\approx\) \(0.6429863756 - 0.1086278092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.156 - 0.987i)T \)
3 \( 1 + (0.0784 + 0.996i)T \)
7 \( 1 + (-0.996 - 0.0784i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.891 + 0.453i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (0.760 - 0.649i)T \)
31 \( 1 + (0.972 + 0.233i)T \)
37 \( 1 + (-0.649 - 0.760i)T \)
41 \( 1 + (-0.760 - 0.649i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.156 + 0.987i)T \)
59 \( 1 + (-0.891 - 0.453i)T \)
61 \( 1 + (-0.233 - 0.972i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.522 + 0.852i)T \)
73 \( 1 + (0.760 - 0.649i)T \)
79 \( 1 + (-0.522 - 0.852i)T \)
83 \( 1 + (0.987 + 0.156i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.233 + 0.972i)T \)
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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

−22.45569681348343428247022075463, −21.58627162948953682448042060577, −20.05912826093421617396978221145, −19.52894838010576336779171861118, −18.85828153548784093570120538606, −18.017945184607415707240968551749, −17.360564528229706295110264946981, −16.67448014428200248202340642319, −15.67545358807487335578555396472, −15.029902373628932973631609079815, −14.03971006279545983832049815322, −13.35924272312939574735973413994, −12.68425859135522472143890175522, −11.97584536559968834361598320594, −10.5033351113914250838337473912, −9.73500417441640661707654360171, −8.72415471521261789527015576378, −8.05976118881367265847000872209, −7.15842425070947117304671473557, −6.46857959126101779017832903999, −5.82351041155005371880912334485, −4.780115491620629384413912612050, −3.464138319387214011554516926979, −2.45689800980857146674878549646, −0.96302835149196906457278924653, 0.24604614517742264277721720892, 2.084637876260008414676089514319, 2.87051435271368317313506181158, 3.934368340635773947773140417984, 4.38104628911188751478883391376, 5.55016005746019418778689925320, 6.58777789499670269849320334713, 7.99491619392200080273555239257, 8.89206588296112835481828304354, 9.55087248050131455036713376593, 10.27510119571661224864138556489, 10.79938098856224470703790224952, 12.063945363495091934808775976368, 12.36498875241293081408079412118, 13.76696618126515260476722723921, 14.11577811373847358689831526356, 15.29484794533748427132936994706, 16.12962186061290841693023561465, 16.95969602235310904727181565741, 17.49525020196580511954069062192, 18.90950198785344433872476379791, 19.24092284946776888051644101509, 20.12962324140141774896653011193, 20.7707199529908867365164340955, 21.68734132226693603572347839887

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