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Properties

Label 2-35-35.19-c2-0-5
Degree $2$
Conductor $35$
Sign $-0.650 + 0.759i$
Analytic cond. $0.953680$
Root an. cond. $0.976565$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.74 − 1.00i)2-s + (−0.784 − 1.35i)3-s + (0.0184 + 0.0319i)4-s + (−4.64 − 1.86i)5-s + 3.15i·6-s + (1.38 − 6.86i)7-s + 7.96i·8-s + (3.26 − 5.66i)9-s + (6.20 + 7.90i)10-s + (4.78 + 8.29i)11-s + (0.0289 − 0.0501i)12-s + 13.9·13-s + (−9.30 + 10.5i)14-s + (1.11 + 7.76i)15-s + (8.07 − 13.9i)16-s + (−8.97 − 15.5i)17-s + ⋯
L(s)  = 1  + (−0.870 − 0.502i)2-s + (−0.261 − 0.453i)3-s + (0.00461 + 0.00798i)4-s + (−0.928 − 0.372i)5-s + 0.525i·6-s + (0.198 − 0.980i)7-s + 0.995i·8-s + (0.363 − 0.628i)9-s + (0.620 + 0.790i)10-s + (0.435 + 0.753i)11-s + (0.00241 − 0.00417i)12-s + 1.07·13-s + (−0.664 + 0.753i)14-s + (0.0740 + 0.517i)15-s + (0.504 − 0.873i)16-s + (−0.528 − 0.914i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.650 + 0.759i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ -0.650 + 0.759i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.217390 - 0.472081i\)
\(L(\frac12)\) \(\approx\) \(0.217390 - 0.472081i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.64 + 1.86i)T \)
7 \( 1 + (-1.38 + 6.86i)T \)
good2 \( 1 + (1.74 + 1.00i)T + (2 + 3.46i)T^{2} \)
3 \( 1 + (0.784 + 1.35i)T + (-4.5 + 7.79i)T^{2} \)
11 \( 1 + (-4.78 - 8.29i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 13.9T + 169T^{2} \)
17 \( 1 + (8.97 + 15.5i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (9.91 + 5.72i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (7.35 + 4.24i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 26.6T + 841T^{2} \)
31 \( 1 + (1.87 - 1.08i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-56.1 - 32.4i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 23.3iT - 1.68e3T^{2} \)
43 \( 1 - 0.506iT - 1.84e3T^{2} \)
47 \( 1 + (34.3 - 59.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-47.7 + 27.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-54.5 + 31.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-43.9 - 25.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (33.7 - 19.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 3.47T + 5.04e3T^{2} \)
73 \( 1 + (12.3 + 21.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (68.8 - 119. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 110.T + 6.88e3T^{2} \)
89 \( 1 + (-84.0 - 48.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 3.28T + 9.40e3T^{2} \)
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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

−16.15938608670835686957847208808, −14.78466862721876780505740933226, −13.30106097118360065251163119191, −11.89128809382544863696241727547, −10.98987150283946229136648696812, −9.581680607046029609172024085236, −8.246427843504957010222082738164, −6.84979120873674651188115454316, −4.35768712333068833700948484371, −0.940112876739732647087778559491, 3.98548720338686912165149214066, 6.27468621053522003210411132324, 8.017467677963602538479346943309, 8.795840042817028390573780579550, 10.48836841791821910279936893106, 11.60492311432591294983126939522, 13.08938941514004482378796051692, 14.94640710104301137840365175301, 15.92182620971160907569881820219, 16.49922343256504493200057887294

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