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Properties

Label 2-935-17.16-c1-0-54
Degree $2$
Conductor $935$
Sign $0.0101 + 0.999i$
Analytic cond. $7.46601$
Root an. cond. $2.73240$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.37·2-s − 2.24i·3-s + 3.66·4-s − i·5-s − 5.34i·6-s − 0.295i·7-s + 3.95·8-s − 2.04·9-s − 2.37i·10-s + i·11-s − 8.22i·12-s + 1.20·13-s − 0.702i·14-s − 2.24·15-s + 2.08·16-s + (−0.0416 − 4.12i)17-s + ⋯
L(s)  = 1  + 1.68·2-s − 1.29i·3-s + 1.83·4-s − 0.447i·5-s − 2.18i·6-s − 0.111i·7-s + 1.39·8-s − 0.680·9-s − 0.752i·10-s + 0.301i·11-s − 2.37i·12-s + 0.333·13-s − 0.187i·14-s − 0.579·15-s + 0.520·16-s + (−0.0101 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0101 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0101 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $0.0101 + 0.999i$
Analytic conductor: \(7.46601\)
Root analytic conductor: \(2.73240\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 935,\ (\ :1/2),\ 0.0101 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.92368 - 2.89429i\)
\(L(\frac12)\) \(\approx\) \(2.92368 - 2.89429i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
11 \( 1 - iT \)
17 \( 1 + (0.0416 + 4.12i)T \)
good2 \( 1 - 2.37T + 2T^{2} \)
3 \( 1 + 2.24iT - 3T^{2} \)
7 \( 1 + 0.295iT - 7T^{2} \)
13 \( 1 - 1.20T + 13T^{2} \)
19 \( 1 - 1.78T + 19T^{2} \)
23 \( 1 - 0.889iT - 23T^{2} \)
29 \( 1 - 0.754iT - 29T^{2} \)
31 \( 1 - 8.99iT - 31T^{2} \)
37 \( 1 - 1.10iT - 37T^{2} \)
41 \( 1 - 6.56iT - 41T^{2} \)
43 \( 1 - 0.439T + 43T^{2} \)
47 \( 1 + 5.78T + 47T^{2} \)
53 \( 1 + 6.57T + 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 1.47iT - 61T^{2} \)
67 \( 1 - 5.55T + 67T^{2} \)
71 \( 1 + 1.91iT - 71T^{2} \)
73 \( 1 - 6.10iT - 73T^{2} \)
79 \( 1 - 4.02iT - 79T^{2} \)
83 \( 1 - 0.817T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 1.27iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.00684193944885477056095781648, −8.819052730660290786048909452139, −7.74976051815089487575513154272, −6.98967837273618779404165662145, −6.40896396123676553588032011040, −5.38146441205478038772954864956, −4.68813138065478554431473651439, −3.48980120974155713272303339731, −2.44441467541495235105029428894, −1.26398642149511854962847417689, 2.28453951942137669685812806614, 3.50362633174842101995944222915, 3.91336287702641271154741538172, 4.85384481884994558912844811406, 5.70958977359698977438797806067, 6.35616951742287670063958050172, 7.48991396771347851548755372491, 8.691031675821647708822237737261, 9.697029510877434751245210281527, 10.53729807350134338396434413304

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