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Properties

Label 2-35-35.34-c4-0-11
Degree $2$
Conductor $35$
Sign $-0.967 + 0.254i$
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  − 7.21i·2-s + 9.48·3-s − 36.0·4-s + (−22.2 − 11.4i)5-s − 68.4i·6-s + (36.4 − 32.7i)7-s + 144. i·8-s + 9·9-s + (−82.3 + 160. i)10-s + 40.4·11-s − 342.·12-s + 222.·13-s + (−236. − 263. i)14-s + (−211. − 108. i)15-s + 467.·16-s + 227.·17-s + ⋯
L(s)  = 1  − 1.80i·2-s + 1.05·3-s − 2.25·4-s + (−0.889 − 0.456i)5-s − 1.90i·6-s + (0.744 − 0.667i)7-s + 2.26i·8-s + 0.111·9-s + (−0.823 + 1.60i)10-s + 0.333·11-s − 2.37·12-s + 1.31·13-s + (−1.20 − 1.34i)14-s + (−0.937 − 0.481i)15-s + 1.82·16-s + 0.786·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.967 + 0.254i$
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: \((2,\ 35,\ (\ :2),\ -0.967 + 0.254i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.202076 - 1.56227i\)
\(L(\frac12)\) \(\approx\) \(0.202076 - 1.56227i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (22.2 + 11.4i)T \)
7 \( 1 + (-36.4 + 32.7i)T \)
good2 \( 1 + 7.21iT - 16T^{2} \)
3 \( 1 - 9.48T + 81T^{2} \)
11 \( 1 - 40.4T + 1.46e4T^{2} \)
13 \( 1 - 222.T + 2.85e4T^{2} \)
17 \( 1 - 227.T + 8.35e4T^{2} \)
19 \( 1 - 180. iT - 1.30e5T^{2} \)
23 \( 1 + 1.00e3iT - 2.79e5T^{2} \)
29 \( 1 - 50.2T + 7.07e5T^{2} \)
31 \( 1 - 1.28e3iT - 9.23e5T^{2} \)
37 \( 1 - 312. iT - 1.87e6T^{2} \)
41 \( 1 - 1.02e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.04e3iT - 3.41e6T^{2} \)
47 \( 1 - 794.T + 4.87e6T^{2} \)
53 \( 1 - 1.04e3iT - 7.89e6T^{2} \)
59 \( 1 + 1.29e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.52e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.83e3iT - 2.01e7T^{2} \)
71 \( 1 + 4.93e3T + 2.54e7T^{2} \)
73 \( 1 + 1.95e3T + 2.83e7T^{2} \)
79 \( 1 - 1.89e3T + 3.89e7T^{2} \)
83 \( 1 + 77.7T + 4.74e7T^{2} \)
89 \( 1 - 320. iT - 6.27e7T^{2} \)
97 \( 1 + 1.26e4T + 8.85e7T^{2} \)
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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

−14.64460468997335495109960075996, −13.82761201396987479683735604095, −12.61991272814063235136874555616, −11.49267812448020210212731984737, −10.46580026447537370884609215307, −8.837416039797746009781394111621, −8.140260623660871182629041564363, −4.38681058252662227993113161865, −3.29111042483137079434970634427, −1.21895894398514212463003854258, 3.77265373106266854191733007917, 5.71660973444130045559930456240, 7.44025004550000723316059872633, 8.267157653843735692526047145099, 9.149598757516792276345261082863, 11.51614637918615723374142967643, 13.51036578751210141804188527334, 14.39314119763098747144140610391, 15.23169782781540083718794063531, 15.79779701386263677456468315420

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