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Properties

Label 2-74-37.14-c2-0-2
Degree $2$
Conductor $74$
Sign $-0.305 + 0.952i$
Analytic cond. $2.01635$
Root an. cond. $1.41998$
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  + (−1.36 + 0.366i)2-s + (−3.77 + 2.18i)3-s + (1.73 − i)4-s + (0.866 + 0.232i)5-s + (4.36 − 4.36i)6-s + (−5.10 − 8.83i)7-s + (−1.99 + 2i)8-s + (5.00 − 8.67i)9-s − 1.26·10-s − 17.7i·11-s + (−4.36 + 7.55i)12-s + (−8.58 − 2.30i)13-s + (10.2 + 10.2i)14-s + (−3.77 + 1.01i)15-s + (1.99 − 3.46i)16-s + (6.79 + 25.3i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−1.25 + 0.726i)3-s + (0.433 − 0.250i)4-s + (0.173 + 0.0464i)5-s + (0.726 − 0.726i)6-s + (−0.728 − 1.26i)7-s + (−0.249 + 0.250i)8-s + (0.556 − 0.963i)9-s − 0.126·10-s − 1.61i·11-s + (−0.363 + 0.629i)12-s + (−0.660 − 0.177i)13-s + (0.728 + 0.728i)14-s + (−0.251 + 0.0674i)15-s + (0.124 − 0.216i)16-s + (0.399 + 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.305 + 0.952i$
Analytic conductor: \(2.01635\)
Root analytic conductor: \(1.41998\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1),\ -0.305 + 0.952i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.148685 - 0.203942i\)
\(L(\frac12)\) \(\approx\) \(0.148685 - 0.203942i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
37 \( 1 + (34.4 + 13.5i)T \)
good3 \( 1 + (3.77 - 2.18i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-0.866 - 0.232i)T + (21.6 + 12.5i)T^{2} \)
7 \( 1 + (5.10 + 8.83i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + 17.7iT - 121T^{2} \)
13 \( 1 + (8.58 + 2.30i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (-6.79 - 25.3i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (20.4 + 5.47i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (2.06 - 2.06i)T - 529iT^{2} \)
29 \( 1 + (11.3 + 11.3i)T + 841iT^{2} \)
31 \( 1 + (-12.3 - 12.3i)T + 961iT^{2} \)
41 \( 1 + (-36.8 + 21.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (26.4 - 26.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 19.4T + 2.20e3T^{2} \)
53 \( 1 + (-18.8 + 32.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-20.4 - 76.1i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-26.6 + 99.5i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-60.8 + 35.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-38.7 - 67.1i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 - 39.1iT - 5.32e3T^{2} \)
79 \( 1 + (38.9 + 10.4i)T + (5.40e3 + 3.12e3i)T^{2} \)
83 \( 1 + (56.1 - 97.2i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-58.1 + 15.5i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-73.8 + 73.8i)T - 9.40e3iT^{2} \)
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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

−14.08800422797062921545615464341, −12.76618863622700172533911634202, −11.30252861294186170631440671517, −10.52693782334469145067860805326, −9.919379619591912004591605844011, −8.262627713059997959569047218888, −6.61378473186945273045355755821, −5.72741674947652815104937331270, −3.89603598002285038353017031363, −0.29678572219178822443850836618, 2.18593451888178028677119272440, 5.19910654086863044525775874156, 6.46518618834093270075516335496, 7.42153144284577347263639498899, 9.233658060494992234341453718250, 10.09574707137396405428470529715, 11.68448873646395220959659218478, 12.19449590876080504247819689760, 12.96575301247273350082707736744, 14.91243894643030978097279591474

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