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Properties

Label 2-35-35.34-c4-0-0
Degree $2$
Conductor $35$
Sign $-0.483 + 0.875i$
Analytic cond. $3.61794$
Root an. cond. $1.90209$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.78i·2-s − 9.48·3-s − 6.93·4-s + (−23.8 − 7.57i)5-s − 45.4i·6-s + (9.59 − 48.0i)7-s + 43.4i·8-s + 9·9-s + (36.2 − 114. i)10-s − 134.·11-s + 65.7·12-s + 54.4·13-s + (230. + 45.9i)14-s + (226. + 71.8i)15-s − 318.·16-s − 411.·17-s + ⋯
L(s)  = 1  + 1.19i·2-s − 1.05·3-s − 0.433·4-s + (−0.953 − 0.302i)5-s − 1.26i·6-s + (0.195 − 0.980i)7-s + 0.678i·8-s + 0.111·9-s + (0.362 − 1.14i)10-s − 1.11·11-s + 0.456·12-s + 0.321·13-s + (1.17 + 0.234i)14-s + (1.00 + 0.319i)15-s − 1.24·16-s − 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.483 + 0.875i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ -0.483 + 0.875i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0426423 - 0.0722808i\)
\(L(\frac12)\) \(\approx\) \(0.0426423 - 0.0722808i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (23.8 + 7.57i)T \)
7 \( 1 + (-9.59 + 48.0i)T \)
good2 \( 1 - 4.78iT - 16T^{2} \)
3 \( 1 + 9.48T + 81T^{2} \)
11 \( 1 + 134.T + 1.46e4T^{2} \)
13 \( 1 - 54.4T + 2.85e4T^{2} \)
17 \( 1 + 411.T + 8.35e4T^{2} \)
19 \( 1 - 561. iT - 1.30e5T^{2} \)
23 \( 1 + 30.3iT - 2.79e5T^{2} \)
29 \( 1 + 66.2T + 7.07e5T^{2} \)
31 \( 1 + 912. iT - 9.23e5T^{2} \)
37 \( 1 + 1.60e3iT - 1.87e6T^{2} \)
41 \( 1 - 241. iT - 2.82e6T^{2} \)
43 \( 1 - 2.13e3iT - 3.41e6T^{2} \)
47 \( 1 + 517.T + 4.87e6T^{2} \)
53 \( 1 - 145. iT - 7.89e6T^{2} \)
59 \( 1 - 4.87e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.94e3iT - 1.38e7T^{2} \)
67 \( 1 - 3.90e3iT - 2.01e7T^{2} \)
71 \( 1 - 428.T + 2.54e7T^{2} \)
73 \( 1 + 8.36e3T + 2.83e7T^{2} \)
79 \( 1 + 4.05e3T + 3.89e7T^{2} \)
83 \( 1 - 630.T + 4.74e7T^{2} \)
89 \( 1 + 1.30e4iT - 6.27e7T^{2} \)
97 \( 1 + 61.6T + 8.85e7T^{2} \)
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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

−16.40726438792160003582832436788, −15.83168864409778276678104882161, −14.55485224329927029859231774836, −13.06301154653639477174475279909, −11.51432505962990727227348863885, −10.70831185747865183448198045390, −8.313702117823554301205648722968, −7.32288903750104086214190115331, −5.92367791064675078628855811911, −4.52744882253376894431762094793, 0.06318840799050324194359853591, 2.74375893231516884113611343042, 4.86484273250298672419406482733, 6.72985181107401930315769546944, 8.729419874812502475160311485630, 10.65858749009371886301927294277, 11.29253698908949059688453832053, 12.05908132518709546897283217309, 13.17963257982650908752365487969, 15.40018826181902660832699383292

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