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Properties

Label 2-350-7.2-c3-0-33
Degree $2$
Conductor $350$
Sign $-0.701 - 0.712i$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (0.5 + 0.866i)3-s + (−1.99 − 3.46i)4-s + 1.99·6-s + (−17.5 − 6.06i)7-s − 7.99·8-s + (13 − 22.5i)9-s + (15 + 25.9i)11-s + (1.99 − 3.46i)12-s − 44·13-s + (−28 + 24.2i)14-s + (−8 + 13.8i)16-s + (−12 − 20.7i)17-s + (−26 − 45.0i)18-s + (−1 + 1.73i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.0962 + 0.166i)3-s + (−0.249 − 0.433i)4-s + 0.136·6-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (0.481 − 0.833i)9-s + (0.411 + 0.712i)11-s + (0.0481 − 0.0833i)12-s − 0.938·13-s + (−0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.171 − 0.296i)17-s + (−0.340 − 0.589i)18-s + (−0.0120 + 0.0209i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1228855922\)
\(L(\frac12)\) \(\approx\) \(0.1228855922\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
5 \( 1 \)
7 \( 1 + (17.5 + 6.06i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (-15 - 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 44T + 2.19e3T^{2} \)
17 \( 1 + (12 + 20.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (91.5 - 158. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 279T + 2.43e4T^{2} \)
31 \( 1 + (-20 - 34.6i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (38 - 65.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 423T + 6.89e4T^{2} \)
43 \( 1 + 305T + 7.95e4T^{2} \)
47 \( 1 + (-228 + 394. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (99 + 171. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-231 - 400. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (140.5 - 243. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (249.5 + 432. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 534T + 3.57e5T^{2} \)
73 \( 1 + (-400 - 692. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-395 + 684. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 597T + 5.71e5T^{2} \)
89 \( 1 + (508.5 - 880. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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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

−10.22065949050031303706110643507, −9.763152719323571150263571216102, −9.071466444022459685936732878020, −7.39431508717148346144092179362, −6.64575161207572844559707476075, −5.36147294597225339212896753575, −4.08919754033454755070771663670, −3.32601400261130615340184937627, −1.77225584633236317638151727199, −0.03539653910385533185494894648, 2.23921325520793022176649181882, 3.58699056727018860217428682364, 4.80002818753950315426870208218, 5.94064346364805967692591943017, 6.78732318148583686839046533342, 7.75333512365454760316741754456, 8.714409588610550305867673500756, 9.696016874468667423521970411652, 10.63717115788903665205994426363, 11.88422089483357715290653118684

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