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Properties

Label 2-170-17.9-c1-0-1
Degree $2$
Conductor $170$
Sign $0.651 - 0.758i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
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  + (−0.707 + 0.707i)2-s + (1.52 + 0.630i)3-s − 1.00i·4-s + (0.382 − 0.923i)5-s + (−1.52 + 0.630i)6-s + (1.24 + 3.01i)7-s + (0.707 + 0.707i)8-s + (−0.198 − 0.198i)9-s + (0.382 + 0.923i)10-s + (3.58 − 1.48i)11-s + (0.630 − 1.52i)12-s + 0.386i·13-s + (−3.01 − 1.24i)14-s + (1.16 − 1.16i)15-s − 1.00·16-s + (−3.76 + 1.68i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.879 + 0.364i)3-s − 0.500i·4-s + (0.171 − 0.413i)5-s + (−0.621 + 0.257i)6-s + (0.471 + 1.13i)7-s + (0.250 + 0.250i)8-s + (−0.0663 − 0.0663i)9-s + (0.121 + 0.292i)10-s + (1.07 − 0.447i)11-s + (0.182 − 0.439i)12-s + 0.107i·13-s + (−0.805 − 0.333i)14-s + (0.301 − 0.301i)15-s − 0.250·16-s + (−0.912 + 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.651 - 0.758i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.651 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11478 + 0.512180i\)
\(L(\frac12)\) \(\approx\) \(1.11478 + 0.512180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (3.76 - 1.68i)T \)
good3 \( 1 + (-1.52 - 0.630i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-1.24 - 3.01i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-3.58 + 1.48i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 0.386iT - 13T^{2} \)
19 \( 1 + (3.89 - 3.89i)T - 19iT^{2} \)
23 \( 1 + (-6.91 + 2.86i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.192 + 0.465i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (10.2 + 4.25i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.12 - 0.464i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.32 + 8.01i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.190 - 0.190i)T + 43iT^{2} \)
47 \( 1 + 5.12iT - 47T^{2} \)
53 \( 1 + (0.609 - 0.609i)T - 53iT^{2} \)
59 \( 1 + (6.03 + 6.03i)T + 59iT^{2} \)
61 \( 1 + (-2.59 - 6.25i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 2.38T + 67T^{2} \)
71 \( 1 + (-8.21 - 3.40i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (2.69 - 6.51i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.55 + 1.88i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (4.21 - 4.21i)T - 83iT^{2} \)
89 \( 1 - 12.0iT - 89T^{2} \)
97 \( 1 + (4.46 - 10.7i)T + (-68.5 - 68.5i)T^{2} \)
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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

−13.00955126905865543750250657011, −11.86431097807590229332680324268, −10.79186802714275153852908894773, −9.268088926660407959346104120062, −8.904421542055094773862466399894, −8.234909293117343806162243636005, −6.58586824546315487296986638963, −5.50161961673844364930241983387, −3.95440901296421645871650771637, −2.10699448507336566543796900045, 1.71167973409309945868722821990, 3.19307696834670773851436173314, 4.57508355598206112097653307860, 6.85382467641243752836354979721, 7.44491635113676832731492789849, 8.733075216133039901066245222327, 9.422421937005211553224148725131, 10.86226907884907046882233217101, 11.23742216124772931335285729357, 12.82809816697372458605918364095

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