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Properties

Label 2-350-175.52-c1-0-19
Degree $2$
Conductor $350$
Sign $-0.873 - 0.486i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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  + (−0.358 − 0.933i)2-s + (0.171 − 3.27i)3-s + (−0.743 + 0.669i)4-s + (1.02 − 1.98i)5-s + (−3.11 + 1.01i)6-s + (−2.48 + 0.912i)7-s + (0.891 + 0.453i)8-s + (−7.70 − 0.809i)9-s + (−2.22 − 0.248i)10-s + (0.237 + 2.26i)11-s + (2.06 + 2.54i)12-s + (−0.0777 − 0.490i)13-s + (1.74 + 1.99i)14-s + (−6.32 − 3.70i)15-s + (0.104 − 0.994i)16-s + (4.93 − 3.20i)17-s + ⋯
L(s)  = 1  + (−0.253 − 0.660i)2-s + (0.0990 − 1.88i)3-s + (−0.371 + 0.334i)4-s + (0.459 − 0.887i)5-s + (−1.27 + 0.413i)6-s + (−0.938 + 0.344i)7-s + (0.315 + 0.160i)8-s + (−2.56 − 0.269i)9-s + (−0.702 − 0.0785i)10-s + (0.0717 + 0.682i)11-s + (0.595 + 0.735i)12-s + (−0.0215 − 0.136i)13-s + (0.465 + 0.532i)14-s + (−1.63 − 0.957i)15-s + (0.0261 − 0.248i)16-s + (1.19 − 0.776i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.873 - 0.486i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.873 - 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242001 + 0.931612i\)
\(L(\frac12)\) \(\approx\) \(0.242001 + 0.931612i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.358 + 0.933i)T \)
5 \( 1 + (-1.02 + 1.98i)T \)
7 \( 1 + (2.48 - 0.912i)T \)
good3 \( 1 + (-0.171 + 3.27i)T + (-2.98 - 0.313i)T^{2} \)
11 \( 1 + (-0.237 - 2.26i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (0.0777 + 0.490i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-4.93 + 3.20i)T + (6.91 - 15.5i)T^{2} \)
19 \( 1 + (-0.929 + 1.03i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (-4.72 + 1.81i)T + (17.0 - 15.3i)T^{2} \)
29 \( 1 + (7.08 + 2.30i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.399 + 1.88i)T + (-28.3 - 12.6i)T^{2} \)
37 \( 1 + (-3.14 + 2.54i)T + (7.69 - 36.1i)T^{2} \)
41 \( 1 + (2.24 - 3.08i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (5.89 - 5.89i)T - 43iT^{2} \)
47 \( 1 + (1.07 - 1.64i)T + (-19.1 - 42.9i)T^{2} \)
53 \( 1 + (-7.90 - 0.414i)T + (52.7 + 5.54i)T^{2} \)
59 \( 1 + (5.31 - 2.36i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (-4.69 + 10.5i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (7.06 + 10.8i)T + (-27.2 + 61.2i)T^{2} \)
71 \( 1 + (-0.335 + 1.03i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-9.07 + 11.2i)T + (-15.1 - 71.4i)T^{2} \)
79 \( 1 + (-3.56 - 16.7i)T + (-72.1 + 32.1i)T^{2} \)
83 \( 1 + (1.74 - 3.42i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (-0.0651 - 0.0289i)T + (59.5 + 66.1i)T^{2} \)
97 \( 1 + (5.00 + 9.82i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.34082639908862332787141877080, −9.729970679970159055014928336341, −9.172904756613849437534865600510, −8.096182981505276882911140221302, −7.27321615268539975197954166762, −6.20797315527501728579405266864, −5.18561221004388721702493320047, −3.10952900923010405662334124993, −2.00920720281360493251163469682, −0.72283969757064814750544442828, 3.19344029225773168069187161519, 3.78767302803217656405550204845, 5.37197007519048248255579698310, 5.98404116572512214631125463509, 7.22608719392157885703030370614, 8.604781650430342705804237674962, 9.414929632808449161064768943586, 10.15697618161851498380886850100, 10.57329579854942716348392180914, 11.58286700366179870031397698231

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